Persistent Private Information Revisited
A

W. B

†

R. V

K

‡

B

S

§

April 29, 2023

Abstract
This paper revisits Williams’ (2011) (henceforth PPI’s) continuous-time principalagent model of optimal dynamic insurance with persistent private information. We identify
three independent issues in PPI that implicate its characterizations of incentive compatible
and optimal contracts: (i) the agent cannot over-report increments of his type, a constraintthat does not follow from the common assumption that the agent cannot over-report his
type; (ii) the agent’s feasible set of reporting strategies does not include standard “no Ponzi”
constraints, without which PPI’s main analysis of infinite-horizon incentive compatibility
is incomplete; and (iii) most importantly, in PPI’s main application, which concerns
hidden endowments, the contract identified as optimal is generically strictly suboptimal.
For this application, we address the three issues by analyzing a class of “self-insurance
contracts” that can be implemented as consumption-saving problems for the agent, andwhich includes the contract derived in PPI as a particular case. We characterize the optimal
self-insurance contract and show that, generically, it strictly dominates PPI’s. Our analysis
does not support PPI’s main economic finding that immiseration generally fails or its
attribution of this failure to continuous time and persistence.

�. Introduction

In an influential paper, Williams (2011) (henceforth PPI) introduces a continuous-timeframework to study optimal contracts in dynamic principal-agent settings where theFirst version: November 2020. Parts of this paper build on and subsume portions of the working
papers Strulovici (2011) and Bloedel, Krishna, and Leukhina (2018). We thank Jaköa CvitaniÊ, Arash
Fahim, Ilya Segal, Neng Wang, and various seminar and conference audiences for helpful comments. We
are particularly grateful to the Co-Editor and three anonymous referees for detailed feedback that greatly
improved the paper. Bloedel is grateful to Caltech for its hospitality and financial support while part ofthis research was conducted, and to the B.F. Haley and E.S. Shaw Fellowship for Economics for financial
support through a grant to the Stanford Institute for Economic Policy Research.
† UCLA. Email: abloedel@econ.ucla.edu‡ Florida State University. Email: rvk3570@gmail.com§ Northwestern University. Email: b-strulovici@northwestern.edu

1agent’s private information is persistent. In the present paper, we identify and address
three independent issues in PPI’s model formulation and analysis, elucidate their consequences for PPI’s conclusions, and discuss implications for the broader literature. In
doing so, we develop some new results and methods that may be of independent interestand highlight some open questions that we hope will stimulate future work.
Overview of PPI. PPI aims to make two methodological contributions and one economically substantive contribution (see pp. 1233–35). 1
The methodological contributions are, first, to formulate a first-order approach toincentive compatibility (IC) in contracting models with a persistent private state (§§2–3)
and, second, to provide a set of sufficient conditions under which that approach is valid(§4). Many of these methods have been fruitfully applied in the subsequent literature toshed light on new and difficult problems in contract theory. 2
For the substantive contribution, PPI applies these techniques to study optimal contracts, formulating the principal’s optimization problem in an abstract setting (§5) and
explicitly solving for optimal contracts in a canonical class of dynamic insurance problems in which the agent’s private information concerns his endowment (§§6–7) or taste
shocks (§8). PPI’s analysis appears to overturn common wisdom about the qualitative
features of dynamic insurance contracts, which are the subject of longstanding literaturesin both micro- and macroeconomic theory. 3 Specifically, in both applications, PPI’s key
finding is that the optimal contract sends the agent to bliss: the agent’s consumption andutility converge almost surely to their upper bounds. This finding goes against the classic
literature’s hallmark result that optimal insurance contracts generate immiseration: theagent’s consumption and utility converge almost surely to their lower bounds. As theclassic literature focuses on discrete-time models with i.i.d. private information, PPI
attributes this discrepancy to fundamental differences in the agent’s IC constraints driven
by continuous time and persistence (e.g., see p. 1235 and pp. 1256–58).
Three Issues with PPI. PPI’s model formulation and analysis contain three issues that
implicate most aspects of the paper’s latter two contributions (sufficient conditions for
IC and analysis of optimal contracts). 4 We summarize these issues below. We list them
1 Throughout, we use page numbers and the § symbol to refer, respectively, to pages and sections in
PPI. We reserve the term “section” for reference to sections in the present paper.
2 For instance, Chen (2021), Cisternas (2017), DeMarzo and Sannikov (2016), Prat and Jovanovic(2014), Ramos and Sadzik (2019), and Sannikov (2014) adopt versions of PPI’s first-order approach andaspects of its proof strategy for establishing full incentive compatibility.
3 See Green (1987), Thomas and Worrall (1990), Atkeson and Lucas (1992), and Phelan (1998) for
classic contributions, Kocherlakota (2010) and Golosov, Tsyvinski, and Werquin (2016) for surveys, andBloedel, Krishna, and Leukhina (2021) for a recent contribution and additional references.
4 These issues do not have any notable implications for PPI’s first contribution (the first-order approach
to IC). Meanwhile, PPI’s economic conclusions about the failure of immiseration and the associated roles2

in their logically most linear order, but note that the third issue is the most substantive.
1. Sign restrictions on misreports: PPI’s description of the model includes the standard
assumption that the agent can only under-report his type. However, PPI’s actual
analysis uses the significantly stronger restriction that the agent can only underreport each increment of his type, implying that the agent cannot correct for any
past under-reports. 5 This restriction, which does not follow from the initial underreporting assumption (Observation 1), is conceptually problematic and has no known
analogues in the literature. Without it, however, the class of IC contracts shrinks andthe analysis of IC requires different arguments. In particular, the proof of PPI’s generalsufficient conditions for IC (Theorem 4.1, p. 1247 and §A.2) and the verification of
IC in PPI’s main application (§6 and §A.3.1) rely on this stronger restriction.
2. Tail restrictions on misreports: PPI initially formulates the agent’s reporting problem
in a finite-horizon setting (§§2–4). However, the entire formulation and analysis
of optimal contracts takes place in an infinite-horizon setting (§§5–8), informally
motivated as a limit of finite-horizon models. In this infinite-horizon setting, PPI omits
standard “no Ponzi” restrictions on the asymptotic growth rate of the agent’s feasible
reporting strategies and does not check whether the agent’s value function satisfies
the appropriate transversality condition. PPI’s main analysis of infinite-horizon IC
is incomplete as a result of these omissions. In particular, the contract identified
as optimal in PPI’s main application (defined below as Contract PPI) genericallyviolates IC under PPI’s assumptions (Observation 2).
3. Generic suboptimality of the contract identified in PPI: PPI’s main application
concerns a risk-sharing model in which the agent has a privately observed endowment(§§6–7). 6 Due to an incorrect numerical characterization of the optimal contract’s
initial condition (see Appendix A for details), the contract that PPI identifies as
optimal (Contract PPI) is, in fact, strictly suboptimal in the generic case that theagent’s endowment has non-zero mean-reversion (Observation 3). Consequently,
PPI’s main economic conclusions about the failure of immiseration and the associatedroles of continuous time and persistence do not follow from that paper’s analysis.
Closely related results in the recent literature further suggest that these conclusions
are, in fact, incorrect (see Section 7.2).
These three issues are logically independent of each other: even if any two are addressed,of continuous-time and persistence, which we show are not warranted, have been informally echoed in theliterature (e.g., Zhang (2009, pp. 637,652); Kapi ka (2013, p. 1029); Prat and Jovanovic (2014, p. 885)).
5 In particular, this restriction prevents the agent from engaging in “one-shot deviations” that consistof under-reporting his type for a small amount of time and truthfully reporting his type thereafter.
6 PPI’s other application (to a taste shock model in §8) considers only the special case in which theagent’s type has exactly zero mean-reversion and is essentially equivalent (modulo a change of variables)
to the analogous special case of the main hidden endowment application. Thus, our analysis effectively
covers both of PPI’s applications.

3the third remains problematic. They are also essentially unrelated to PPI’s continuoustime formulation: the same observations apply almost verbatim to the natural discretetime versions of PPI’s model and Contract PPI. The generic suboptimality of Contract
PPI—which we view as the most important and economically substantive issue—remains
an issue regardless of the sign or tail restrictions that one imposes on the agent’s feasible
set, and independently of whether one formulates the model in continuous or discrete
time. We study the first two issues primarily to emphasize this point.
Addressing the Issues. After recalling PPI’s general model (Section 2) and main
application (Section 3), we describe the three issues summarized above (Section 4).
These issues implicate the general analysis in PPI. For our main analysis (Sections 5
and 6), we then turn our attention to PPI’s primary application, in which the agent has a
privately observed endowment that follows an OU process (i.e., continuous-time version
of a Gaussian AR(1) process) and CARA utility over consumption.
In this setting, we address the first two issues by (i) allowing for a range of sign
restrictions on the agent’s feasible misreports (including arbitrary over- and underreporting) and (ii) imposing a standard “no Ponzi” constraint on the agent that limits
the asymptotic growth rate of his misreports. This allows us to demonstrate that (a)under our “no Ponzi” constraint, Contract PPI is IC given any sign restrictions on theagent’s misreports, and (b) our “no Ponzi” constraint on the agent is a minimal sufficientcondition for Contract PPI to be IC. We then elucidate the third issue by showing thatContract PPI is strictly suboptimal in the generic case that the agent’s endowment has
non-zero mean-reversion. We approach the suboptimality issue in two steps:
1. We first observe that the agent’s consumption under Contract PPI is characterized
by a standard Euler equation, and therefore coincides with the solution to a standard
consumption-saving problem for the agent, in which there is no principal and theagent simply self-insures by investing in a risk-free bond at the ambient market rate
(which is equal to the discount rate). Consequently, PPI’s main economic results,
which concern the long-run properties and comparative statics of Contract PPI, can beinterpreted as standard results about precautionary savings in self-insurance problems.
Notably, the classic contracting literature has found that optimal contracts typicallydo not coincide with solutions to self-insurance problems (e.g., Allen (1985); Coleand Kocherlakota (2001)).
2. We then introduce a new class of Self-Insurance Contracts (SI Contracts) definedin terms of an indirect implementation in which the principal acts as the agent’s“bank.” 7 Specifically, the principal provides the agent with some initial wealth and7 As we discuss in Section 5.1, versions of our solution to the agent’s self-insurance problem, whichin our setting arises as his best-response within an SI Contract, appear in the self-insurance literature(Caballero 1990; Wang 2003, 2006). To the best of our knowledge, neither our definition nor our analysis
of optimality for SI Contracts (Section 5.2–5.4) has close analogues in that literature.

4then allows the agent to self-insure at a risk-free interest rate equal to the sum of
the ambient market rate and a savings tax that the principal charges the agent. As
suggested by Step 1, Contract PPI is the specific SI Contract with zero taxes. However,whenever the agent’s endowment has non-zero mean-reversion, we show that the
optimal SI Contract imposes strictly positive taxes. We interpret this result in termsof the agent’s precautionary savings behavior.
We also derive some additional properties of SI Contracts that may be of independentinterest beyond this paper.
In Section 6, we provide two proofs that all SI Contracts, including Contract PPI,
are IC when re-formulated as direct mechanisms: one based on an “indirect” revelation
principle argument and the other based on a “direct” analysis of the agent’s reporting
problem. 8 Each approach is independently instructive, requires different analysis thanthat in PPI, reveals similarities between IC in continuous- and discrete-time models, andmay be useful for analyzing IC in other continuous-time contracting models.
(Fully) Optimal Contracts. Our analysis of SI Contracts raises two questions:
1. Are there conditions under which Contract PPI is, in fact, optimal? We provide two
positive answers. First, it is optimal in an alternative model in which the agent cancovertly save and borrow at the market rate (“hidden savings”), linking PPI’s analysis
to classic discrete-time studies of optimal hidden savings contracts (Allen 1985; Coleand Kocherlakota 2001). Second, it is optimal in the non-generic case of PPI’s originalmodel in which the agent’s endowment has exactly zero mean-reversion (“permanent
shocks”), confirming PPI’s finding in this special case. We thus obtain optimality
foundations for Contract PPI, albeit in distinct and rather specific environments. 9
2. Is the optimal SI Contract “fully” optimal (i.e., among all IC contracts)? We offera negative answer: under regularity conditions, the fully optimal contract strictly
dominates the optimal SI Contract whenever the agent’s endowment has non-zeromean-reversion (“transient shocks”). While our analysis of SI Contracts has some implications for the fully optimal contract under transient shocks, a full characterizationof the latter contract remains an important open problem.
We summarize these analyses in Sections 5.4 and 7 (details are in Appendices H and J).
8 That is, for the “indirect” approach, we construct an explicit mapping between consumption-saving
strategies in the indirect mechanism and reporting strategies in the direct mechanism. One implication ofthis mapping is that our “no Ponzi” restriction on reporting strategies is equivalent to the standard no
Ponzi condition on savings in the agent’s self-insurance problem.
9 The assumption of hidden savings, while realistic in some settings, corresponds to a fundamentallydifferent model than that studied in PPI and much of the social insurance literature, which focuses on theimplications of the agent’s private information in isolation. The assumption of permanent endowment
shocks is knife-edge and constitutes a fundamental departure from the classic literature with i.i.d. types:
as we discuss in Section 7, it leads to qualitatively different tradeoffs for the principal than arise undertype processes with even arbitrarily slow mean-reversion.

5

Broader Implications. In Section 7, we discuss implications of our analysis for theinterpretation of PPI’s results and for the broader literature.
• Immiseration and Persistence: PPI’s main economic conclusion that immiseration
fails under persistent private information is based on the analysis of Contract PPI.
We find that this conclusion is warranted only in the non-generic case of permanent
shocks; it is driven by the absence of mean-reversion in the agent’s type process—rather than persistence per se or continuous time, as asserted in PPI. We further argue
that immiseration should, in fact, be expected to hold under the optimal contract in
PPI’s model when shocks are transient.
• Continuous vs. Discrete Time: A central claim in PPI is that its results differ fromthose in the prior discrete-time literature because IC constraints are qualitatively
different in discrete- and continuous-time models. Our analysis casts doubt on thisclaim: our SI Contracts (including Contract PPI) have precise discrete-time analogues,
and our analysis of IC reveals fundamental parallels between incentive constraints incontinuous- and discrete-time settings.
�. ModelSection 2.1 introduces PPI’s general model. Section 2.2 introduces various possible
restrictions on the agent’s feasible set of reporting strategies.
2.1. Environment
Time is continuous and runs over an infinite horizon. At t “ 0, a risk-neutral principal(she) offers an insurance contract to a risk-averse agent (he). Once the contract is signed,
neither party may renege at a later date.
Type Process. At each instant t, the agent privately observes his type, bt P R. The
agent’s type process b “ pbt qt•0 evolves according to the equation[2.1]

dbt “ pµ ´ bt q dt ` dWt ,where ° 0 and W “ pWt qt•0 is a standard Brownian motion. The initial condition
b0 P R is common knowledge. The parameter • 0 specifies the rate of mean reversion.
When “ 0, b is a Brownian motion with constant drift. We refer to this as the permanent
shock case because the time-t shock (the Brownian increment dWt ) has a non-vanishingadditive effect on bT for all T ° t. When ° 0, b is an Ornstein-Uhlenbeck (OU) process.
We refer to this as the transient shock case because the time-t shock has a vanishingeffect on bT as T Ñ 8. Note that smaller values of correspond to greater persistence. 10
10Given any realization of bt and time T ° t, the solution to [2.1] is bT “ µ{ ` pbt ´ µ{ qe´ pT ´tq `≥Te´ pT ´tq t´ pT ´tq
Te ⌧ dW⌧ . It follows that db. In the language of Pavan, Segal, and Toikka (2014),
dbt “ e6

We let P denote the probability measure over paths of b. 11
Reporting Strategies. At each instant t, the agent reports a type yt P R. The processy “ pyt qt•0 is the agent’s reporting strategy, and is assumed to be adapted to the filtrationof process b (henceforth, b-adapted). 12 The process m “ pmt qt•0 where mt :“ yt ´ bt
is the agent’s misreporting strategy, and is also b-adapted. Thus, mt ° 0 corresponds
to over-reporting one’s type while mt † 0 corresponds to under-reporting it. Clearly,
each reporting strategy y uniquely defines a corresponding misreporting strategy mand vice versa. However, it is useful to distinguish between these objects because theprincipal only observes the realized sample path of y , while the agent observes therealized sample path of b and his own misreporting strategy. Every misreporting strategy
m induces a probability measure Pm over paths of y . We let y ˚ :“ b and m˚ :“ 0 denote˚
the truthful reporting and misreporting strategies, respectively, and let P˚ :“ Pm denotethe corresponding measure over report paths. Note that while P˚ coincides with P, they
are measures over paths of different processes (y and b, respectively).
As in PPI, we assume that the agent’s misreports have absolutely continuous≥tsample paths, i.e., there exists a process “ p t qt•0 such that mt ” 0 s ds (where “””denotes that P-a.s., the processes are equal for almost all t • 0). Thus, the agent’s report
evolves as dyt “ dbt ` t dt, where the drift adjustment t corresponds to misreporting
the increment dbt . Let M denote the space of such misreporting strategies.
Contracts. A contract is a continuous y -adapted process s “ pst qt•0 that specifies
transfers (of the consumption good) from the principal to the agent as a function of thehistory of reports.
Agent’s Incentives. The agent’s type determines his preference over consumption: if
the agent’s current type is bt and he receives transfer st P R from the principal, his flow
utility is vpst , bt q. The agent may be restricted to a subset of misreporting strategies, his
feasible set F Ñ M, that contains the truthful strategy m˚ . Given the agent’s feasible set
e´ pT ´tq is the impulse response of bT to bt . Smaller values ofwhen “ 0, the impulse responses are identically 1.

correspond to larger impulse responses;

11Formally, this requires specifying the measure space of sample paths. Following PPI (p. 1239), we use
the space of continuous paths Cpr0, 8qq endowed with the standard Borel sigma-algebra (cf. Appendix C).
12That is, yt is measurable with respect to the sigma-algebra generated by the paths pb⌧ q⌧ Pr0,ts . PPI
further assumes that y is predictable with respect to the filtration generated by b (i.e., is b-adapted and,moreover, yt does not depend on the contemporaneous type bt ). However, this distinction is inconsequential
under PPI’s assumption, which we adopt below, that the misreporting process m :“ y ´ b has absolutely
continuous sample paths. (When y has continuous paths, it is adapted if and only if it is predictable.)An analogous qualifier is relevant when we define contracts below as being y -adapted, rather than y predictable. We assume mere adaptedness here because it is the appropriate assumption when, in Section 6,
we consider extending the agent’s strategy space to allow for discontinuous “jump” reports (see Section 6,
Section 7.3, and Appendix I for further discussion).

7F , a contract is said to be F -incentive compatible (F -IC) if it satisfies 13
„ª 8
⇢
˚
m
´⇢t
[IC]
m P arg max E0
e vpst , bt q dt ,mPF

0where ⇢ ° 0 is the agent’s discount rate.

Principal’s Problem. The principal also has discount rate ⇢ ° 0. A standard interpretation, which will be important later, is that ⇢ represents the interest rate at which theprincipal finances the contract on a risk-free bond market. Given the agent’s feasible
set F Ñ M, the principal chooses a contract to minimize the expected lifetime cost oftransfers to the agent (under truthful reporting)
E˚0

[2.2]

„ª 8

e

´⇢t

0

st dt

⇢

subject to the contract (i) being F -IC and (ii) satisfying the promise keeping constraint“≥8‰
q0 § E˚0 0 e´⇢t vpst , bt q dt , where the initial promised utility q0 † 0 is a given parameter.
An F -optimal (full-commitment) contract is any contract that minimizes the principal’s
costs subject to these two constraints. 14
Remark 2.1. The model described above is identical to the model introduced in PPI,
except for two distinctions:(i) We define the incentive compatibility and optimality of a contract as a function of
the agent’s feasible set F Ñ M. This allows us to be more explicit than PPI about
which restrictions are being imposed on the agent’s strategy space (see Section 2.2
below) and their implications for the class of IC contracts (Section 4).
(ii) We formulate the model directly over an infinite horizon. Although PPI initially formulates a finite-horizon version of the model (§2) and analyzes incentive compatibility
in that context (§§3–4), the entire analysis of optimal contracts, both in an abstract
setting (§5) and in applications (§§6–8), takes place in a informal infinite-horizon
limit of that initial finite-horizon model (see Footnote 23 below). We proceed directly
to the infinite-horizon formulation because (a) following PPI, it is the relevant setting
for studying optimal contracts, and (b) PPI’s infinite-horizon model as written is not
fully specified because it relies simultaneously on a finite-horizon formulation of
the agent’s reporting problem and an infinite-horizon formulation of the principal’scontracting problem (see Section 4.2 below).
13We let Emt denote the expectation with respect to the agent’s time-t information under misreporting˚

strategy m P M. We also let E˚t :“ Emfor brevity, as in [2.2] below.
t
14As usual, we also implicitly restrict attention to the class of contracts for which [2.2] is well-defined.

8

2.2. Restrictions on Reporting Strategies
We introduce various restrictions on the agent’s feasible set F , some of which come from
PPI and some of which are new. Doing so allows us to clarify the restrictions imposed in
PPI, and to modify these restrictions after showing that some of them are problematic.
Throughout, it will be useful to keep in mind the following elementary fact:
Fact 1. If the feasible sets F, F 1 Ñ M satisfy F Ñ F 1 , then:(i) Every F 1 -IC contract is also F -IC, i.e., the set of F 1 -IC contracts is smaller than theset of F -IC contracts.
(ii) The optimal F 1 -IC contract has a weakly higher cost to the principal (i.e., is no better)
than the optimal F -IC contract.
Sign Restrictions. A class of restrictions considered in PPI concerns the sign of theagent’s misreports. One such restriction is that the agent can only under-report his type,so that his misreporting process is everywhere non-positive. For reasons discussed in
Section 4.1 below, we dub this restriction No Hidden Borrowing (NHB) and denote theset of misreporting strategies consistent with NHB by[NHB]

M´ :“ tm P M : m § 0u .

A stronger restriction is that the agent can only under-report the increments of his type, sothat the process is everywhere non-positive. Because this implies that the misreporting
process has non-increasing sample paths, we refer to it as Increasing Magnitude of Lies
(IML) and denote the set of misreporting strategies consistent with IML by[IML]

M§ :“ tm P M :§ 0u .

As explained below in Section 4.1, PPI motivates NHB as part of the model formulation
but relies on IML for the formal analysis.
Absolute Continuity of Measures. Another class of restrictions considered in PPIconcerns the kinds of misreporting strategies that are (un)detectable by the principal.
Recall that each m P M formally induces a measure Pm over sample paths of y ,˚with P˚ :“ Pm the measure induced by truthtelling. PPI assumes, as part of the finitehorizon model formulation (pp. 1240–42), that the agent can only misreport in waysthat generate an absolutely continuous (AC) change-of-measure: given the finite horizonm
r0, T s and letting Pm
T denote the marginal of P over r0, T s-truncated sample paths, PPI
˚m˚
assumes that m is feasible only if Pm
T is AC with respect to PT (denoted PT ! PT ). 15
0
0
15To be precise, PPI assumes that Pm
T ! PT , where P is the measure induced by the non-truthfulstrategy under which y is a driftless Brownian motion. For any finite T , this is equivalent to requiring

9Economically, this restriction is meant to capture the fact that, in continuous time,
there are certain kinds of misreports that the principal can instantaneously detect (withprobability one), and therefore deter at zero expected cost by “shooting the agent” upondetection (cf. PPI p. 1240). 16 Technically, it facilitates PPI’s use of Girsanov’s Theorem
to reformulate the agent’s reporting problem as one of choosing a “density process” forthe change-of-measure (§2.3).
With an infinite horizon, however, there are two standard but distinct notions of
AC changes-of-measure. Specifically:
˚• Pm is locally AC with respect to P˚ if the finite-horizon marginals satisfy Pm
t ! Ptfor all t ° 0. We let[LAC]

˚
MLAC :“ tm P M : Pm
t ! Pt for all t ° 0u

denote the class of strategies inducing locally AC changes-of-measure. Intuitively,
no m P MLAC is detectable by the principal in finite time.
• Pm is globally AC with respect to P˚ if Pm ! P˚ , i.e., the measures over entire
infinite-horizon paths are AC. We let[GAC]

MGAC :“ tm P M : Pm ! P˚ udenote the class of strategies inducing globally AC changes-of-measure. Intuitively,
no m P MGAC is detectable even in infinite time (i.e., “at t “ 8”, after the entire
sample path has been observed).
It is well known that GAC is strictly more demanding than LAC because the former
imposes fairly strong restrictions on the asymptotic behavior of the m process as t Ñ 8(see Appendix C). As we discuss in Section 4.2, PPI does not explicitly specify whichnotion is meant to be imposed in the infinite-horizon model.
No Ponzi Constraints. Our analysis will illustrate the importance of imposing tailrestrictions that constrain the asymptotic growth rate of the agent’s misreports (see˚m0that Pm
T ! PT . However, in the infinite-horizon model, requiring that P ! P would imply that truthfulreporting is infeasible, which would be inappropriate (cf. [GAC] below).
16The AC change-of-measure assumption is stronger than this intuition suggests. Formally, suppose theprincipal fixes a P˚ -null event N Ä Cr0, 8q of “detectable lies” (e.g., sample paths that have different
quadratic variation than b) and “shoots” the agent if and only if he reports a path in N . While this constrains
the agent to strategies m for which Pm pN q “ 0, he could still set Pm pN 1 q ° 0 for a different P˚ -null set
N 1 . For instance, he could pick any path ŷ P Cr0, 8qzN and always report it; the principal, who only
observes one path of reports, would be unable to distinguish this strategy from truthtelling conditionalon ŷ being the realized path of b. Nonetheless, the AC change-of-measure assumption rules out suchstrategies by preventing the agent from placing positive weight on any P˚ -null event. Which misreports
should be deemed detectable is a subtle open question for the continuous-time contracting literature (cf.
Acciaio, Crowell, and CvitaniÊ (2022)).

10

Section 4.2). However, we find it convenient to impose tail restrictions that are weaker—
and admit an arguably more natural economic interpretation—than those implied by
GAC, and that, unlike GAC, can be formulated as pathwise constraints. To this end, we
say that misreporting strategy m satisfies the no Ponzi condition (at rate r ° 0) if[NP-m]

lim e

tÑ8

´rt

ªt
0

m⌧ d⌧ • 0

P-a.s.

Notice that [NP-m] constrains the asymptotic growth rate of the agent’s under-reports
only by ruling out sample paths along which mt diverges to ´8 too quickly relative
to the rate r. As formalized in Section 6, [NP-m] is analogous to the standard no Ponzi
constraint limtÑ8 e´rt At • 0 that arises in a consumption-saving problem with “interest≥trate” r ° 0 and “asset process” At ” 0 m⌧ d⌧ . We denote the set of misreporting
strategies consistent with this no Ponzi constraint by[2.3]

Mr :“ tm P M : [NP-m] holds for rate ru .

Clearly, [NP-m] is strictly more permissive for higher rates: Mr à Mr whenever r † r1 .
Moreover, GAC implies that [NP-m] holds for all rates r ° 0 (see Appendix C).
1

Notation. We define M´r :“ Mr X M´ and M§r :“ Mr X M§ , with a similar convention
for M´LAC , M§LAC , and so on. We also adopt the convention that “NHB” stands for “theassumption that the agent’s feasible set is F “ M´ ,” and so on.
�. Hidden Endowment Application

In this section, we recall the setting of PPI’s main application and the contract derived
as optimal therein (§§6–7 and §A.3).
Setting. The agent’s time-t type bt now corresponds to his endowment at that time. The
agent’s utility of consumption at time t is given by vpst , bt q “ upst ` bt q, where u takes
the CARA form upcq “ ´e´✓c for some ✓ ° 0. Given this structure, it is convenient tore-express contractual variables as follows. Define the agent’s y -adapted (recommended)consumption process c “ pct qt•0 by ct :“ st ` yt and his y -adapted (recommended)flow utility process u “ put qt•0 by ut :“ upct q. These are the consumption and flow
utility processes intended by the contract, presuming that the agent is truthful. Thus,
the agent’s actual consumption is given by the b-adapted process cm “ pcmt qt•0 wheremct :“ ct ´ mt “ st ` bt . In this context, we can interpret mt as the amount that the agent“diverts” for private consumption.
Contract PPI. To describe the contract identified as optimal in PPI—hereafter Contract PPI—we introduce three processes. First, define the y -adapted process W y “≥tpWty qt•0 by Wty :“ yt ´ b0 ´ 0 pµ ´ y⌧ q d⌧ . That is, W y is the shock process that the11principal would infer the agent faced if (a) the principal were to assume that the agent is
truthful and (b) the agent actually follows strategy y . Note that W y coincides with thestandard Brownian motion W when the agent follows the truthful strategy y ˚ and is a
Brownian motion with drift more generally (viz., whenever m P MLAC ). Second, define
the y -adapted promised utility process q “ pqt qt•0 by[3.1]

qt :“ E˚t„ª 8e´⇢p⌧ ´tq⇢u⌧ d⌧ .

tThus, qt is the agent’s lifetime utility from time t onward under truthful reporting. Third,
define the y -adapted (negative) marginal promised utility process p “ ppt qt•0 by[3.2]

pt :“ E˚t„ª 8e

´p⇢` qp⌧ ´tq⇢✓u⌧ d⌧ .

t

Recall that ✓u⌧ ” ´u1 pc⌧ q for CARA utility. Therefore, pt represents the agent’s “marginalincentives” for misreporting by a small amount at time t conditional on having reported
truthfully at all dates ⌧ † t. In particular, [3.2] is the continuous-time version of the
dynamic envelope formula (Pavan, Segal, and Toikka (2014, Theorem 1)) and we caninformally view pt as the derivative dqt {dbt , which is a local measure of the agent’s
on-path information rents. PPI argues that IC contracts can be written recursively with
qt and pt as state variables (pp. 1244–46).
Definition 3.1 (Contract PPI). The contract identified as being (uniquely) optimal in
PPI (henceforth Contract PPI) is that under which promised utility satisfies
[3.3]

`
˘
qt “ q0 exp ´ 12 pk0˚ q2 t ´ k0˚ Wty ,marginal promised utility satisfies pt “ k0˚ qt , and recommended consumption is[3.4]

ct “ cpq0 , ⇢q `pk0˚ q2k˚t ` 0 Wty ,2✓✓where k0˚ :“ ⇢✓{p⇢ ` q and c̄pq, rq :“ ´ logp´rqq{✓.
Contract PPI has several striking features that will be important going forward.
Fact 2. Under truthful reporting, Contract PPI satisfies the following properties:(i) It generates long-run bliss: As t Ñ 8, we have qt , ut Ñ 0 and ct Ñ `8 almost surely,i.e., the agent receives maximal utility and consumption in the long-run. 17
17The fact that ct Ñ `8 a.s. follows from the Strong Law of Large Numbers for Brownian Motion
applied to [3.4] (with truthful reporting, y “ y ˚ ), and implies the asserted a.s. convergence results for ut
and qt .

12

(ii) It is Markovian in promised utility: The continuation contract is a function of qt alone
because pt ” k0˚ qt and, substituting [3.3] into [3.4], we have ct ” cpqt , ⇢q.
(iii) Promised utility is a martingale: Applying Itô’s lemma to [3.3] yields dqt “ ´ k0˚ qt dWty .
(iv) It has a constant utility delivery rate of ⇢: Formally, ut ” ⇢qt , which means theprincipal delivers promised utility at a constant rate ⇢.
PPI emphasizes parts (i)–(iii) of Fact 2 (pp. 1253–54) and notes property (iv) (p.
1253, second-to-last display). Each property in Fact 2 stands in contrast to the literature’s
findings in closely related settings:(a) In contrast to Fact 2(i), the literature has found that, when private information is either(i) i.i.d. or (ii) persistent and mean-reverting, optimal insurance contracts generate
immiseration (which in the current notation means qt , ut , ct Ñ ´8). See Section 7.2
below for references and further discussion.
(b) In contrast to Fact 2(ii), the literature has found that, when private information ispersistent, optimal contracts typically cannot be written recursively in promised
utility alone. See Section 7.1 below for references and further discussion.
(c) Points (iii) and (iv) of Fact 2, together with the property of CARA utility that u1 pcq “´✓upcq, imply that the agent’s marginal utility of consumption u1 pct q is a martingale,so that his consumption obeys the Euler equation familiar from consumption-saving
problems. In contrast, the literature has found that optimal contracts typically do
not induce the agent’s Euler equation, unless the agent has access to hidden savings
outside of the contract. See Section 5 below for references and further discussion.
PPI emphasizes feature (a) above (pp. 1235, 1257, 1264) but does not note either (b) or(c). 18 We will see in Section 4.3 that property (c) is key to the observation that Contract
PPI is (generically) suboptimal.
�. Issues in PPIThis section formally describes the three issues in PPI’s model formulation and analysis(with some details in Appendices A and D). Section 4.1 describes how PPI introduces
the NHB assumption that m § 0 as part of the model formulation, but then bases theformal analysis on the strictly stronger IML assumption that § 0 (Observation 1).
Section 4.2 describes how PPI omits tail restrictions (such as GAC or NP-m) on theagent’s feasible set of mispeporting strategies, so that, among other things, Contract PPI
18PPI does emphasize that the inverse Euler equation—i.e., the property that 1{u1 pct q is a martingale,which arises as an optimality condition for the principal in many dynamic Mirrleesian models—is notsatisfied in either of PPI’s applications (pp. 1235-36, 1257-58, 1264). However, this is not relevant forunderstanding PPI’s results. It is well known that the inverse Euler equation holds when the agent’s utility
is additively separable across his consumption and private type, in both discrete and continuous time
(Golosov, Kocherlakota, and Tsyvinski 2003; Zhang 2009; Farhi and Werning 2013; Kapi ka 2013).
Conversely, it typically fails in settings without such separability, including the discrete-time models on
which PPI’s applications are based (Thomas and Worrall 1990; Atkeson and Lucas 1992).

13

Sign Restriction
Tail restriction

IML

NHB

None

None / LAC

Not IC

Not IC

Not IC

NP-m (r † ⇢)IC, suboptimal

GAC

IC, suboptimal IC, suboptimalIC, suboptimalIC, suboptimal IC, suboptimal(only m˚ ” 0 feasible)

Table 1: Properties of Contract PPI under transient shocks ( ° 0) and different restrictions on reporting strategies.
is generically not IC under PPI’s assumptions (Observation 2). Perhaps most importantly,
Section 4.3 shows that Contract PPI is generically strictly suboptimal, even after the firsttwo issues have been addressed (Observation 3).
Table 1 summarizes the implications of these issues for PPI’s hidden endowment
application and Contract PPI, which motivate our analysis in the remainder of the paper.
The first two issues also have implications for PPI’s abstract analysis of IC and optimal
contracts, which we discuss in Sections 4.1 and 4.2 for completeness. Readers interested
primarily in the suboptimality of Contract PPI—which we view as the most importantand economically substantive issue—and the economic implications thereof may proceed
directly to Section 4.3 with little loss of continuity.
4.1. Sign Restrictions on Misreporting Strategies
As part of the model formulation, PPI initially imposes the NHB assumption that onlym P M´ are feasible for the agent, writing (p. 1239): “To simplify matters, I assume
that the agent cannot overreport the true state, so yt § bt [i.e., mt § 0].” This restriction
is natural and common in the literature; as noted in PPI (p. 1239), it captures the ideasthat (i) the agent cannot borrow or save outside of the contract and (ii) the agent’s
endowment is partially verifiable (e.g., the principal can require that he deposit thereported amount in a joint account). However, PPI’s formal analysis is based on thestronger IML assumption that only m P M§ are feasible for the agent. PPI writes (onp. 1240): “Since the agent can report (or deposit) at most his entire state [i.e., mt § 0],
we must have t § 0.” Unfortunately, this assertion is incorrect: NHB (m § 0) is astrictly weaker assumption than IML ( § 0). There exist many non-positive functions
t ﬁÑ mt with locally strictly positive derivatives t “ dmt {dt ° 0. 19 Formally, the set of
19NHB requires that t § 0 if and only if mt “ 0 (“on path”) and allows for any t P R when
mt † 0 (“off path”). Two points warrant clarification. First, PPI defines mt as the “stock of lies” (p. 1240).

14

strategies satisfying IML is strictly smaller than the set satisfying NHB, regardless ofthe tail restrictions that one imposes.
Observation 1. M§: à M´: for all : P R`` Y tLAC, GACu.
Proof. The weak inclusions are trivial, so it suffices to find an m P M´GAC zM§ (GAC is
the strongest tail restriction). Let mt :“ ´t for t P r0, 1q, mt :“ t ´ 2 for t P r1, 2q, andmt :“ 0 for t • 2. Then m P M´ zM§ because mt § 0 everywhere and t “ 1 for t P
r1, 2q, and m P MGAC because is bounded and mt ” 0 for t • 2 (cf. Appendix C).
PPI’s reliance on IML is notable for two reasons: 20(i) If PPI intended to assume only NHB, then the analysis pertaining to the verificationof IC is incomplete because it relies on IML. In PPI’s general model, the distinction
between NHB and IML implicates the sufficient conditions for IC in PPI’s Theorem
4.1, the proof of which (in §A.2) would not go through as stated without IML (seeAppendix D). In the hidden endowment application, this distinction implicates theattempted verification (in §A.3.2) that Contract PPI is IC in the generic ° 0 case,which is not sufficient to show that Contract PPI would be IC under NHB. This
attempted verification also involves the derivation of a value function for the agent(restated as V W in [4.1] below) that (a) diverges to ´8 as the current misreportmt † 0 approaches a finite value M † 0 and (b) is different from the agent’s valuefunction under NHB, which is everywhere finite (see Remark B.2 in Appendix B).
This indicates that IML is a much more stringent constraint on the agent than NHB. 21(ii) If PPI intended to adopt IML as a separate assumption, no economic motivation for it
is given. To the best of our knowledge, it has no analogue in the literature. IML implies
that once the agent under-reports his endowment, he can never revert to truthtelling:mt † 0 implies that m⌧ § mt † 0 for all ⌧ • t. Unlike NHB, this stronger property
does not follow from the principal’s ability to partially verify the agent’s endowment
However, mt is a flow variable with the same units as bt . Second, PPI refers to t “ 0 as “truth-telling”(pp. 1267 and 1272) or corresponding to a “truthful current report” (p. 1244) even at histories wheremt † 0. This terminology blurs the distinction between truthful reporting (mt “ 0) and truthful reportingof increments ( t “ 0), potentially generating confusion about the definition of IC and its implications
for the agent’s behavior at off-path histories (which we describe in Section 6.4 below).
20To our knowledge, references to PPI in the literature only mention the weaker NHB assumption. For
instance, Kapi ka (2013, p. 1029) writes “[PPI] assumes that the agent cannot overstate her true shock[i.e., endowment],” and Battaglini and Lamba (2019, p. 1459, fn. 36) write “[PPI] limits the set of possible
deviations available to the agent (who can report only incomes lower or equal to the true income).”21Intuitively, when mt † 0 the principal expects stronger positive mean-reversion than actually occurs,
and so punishes the agent for not reporting increments dyt ° dbt . Meanwhile, the IML constraint forces
the agent to report increments dyt § dbt , so that he cannot avoid such punishments. As m Œ M , thepunishments become so severe that the agent’s continuation value decreases without bound. As we showin Section 6.4, if the agent were not constrained by IML, he would find it optimal to “immediately” correct
for all past under-reports by “jumping” back to mt “ 0.

15

reports. By construction, it also rules out the possibility of “one-shot” deviations,which are central to virtually all dynamic analyses of incentive compatibility, making
PPI’s model and analysis incomparable to those in the literature.
Our Approach. Given that IML is used throughout PPI’s formal analysis, we henceforth adopt the perspective that IML was PPI’s intended assumption. Even under thisstronger assumption, PPI’s main analysis of infinite-horizon IC omits necessary tailrestrictions on the agent’s feasible set and Contract PPI is generically suboptimal (Observation 2 and 3 below). However, our main analysis (Sections 5–7) imposes neither IML
nor NHB, allowing us to (a) bypass what may be viewed as an economically problematic
assumption and (b) demonstrate that Contract PPI remains generically suboptimal under
any sign restrictions that one might wish to adopt (per Fact 1). To analyze IC without
IML, we employ techniques, explained in Section 6, that may prove relevant in othercontinuous-time contracting models.
4.2. Tail Restrictions on Misreporting Strategies
PPI’s definition and analysis of IC contracts proceeds in two steps. First, PPI formulates
the model over a finite time horizon r0, T s (§2) and then provides necessary conditions(§3) and sufficient conditions (§4) for finite-horizon IC. Second, PPI formulates theprincipal’s problem of finding optimal contracts in an infinite-horizon version of themodel, first in an abstract setting (§5) and then in two solved applications (§§6–8).
There are two standard approaches for carrying out the second step, each of whichis common in the literature and has distinct advantages: 22(i) Use the definition of finite-horizon IC from the first step to solve for optimal contracts
over each finite horizon r0, T s, and then study the limit contract as T Ñ 8.
(ii) Define and characterize IC contracts directly over the infinite horizon r0, 8q, andthen solve for the optimal such contract.
PPI adopts neither of these approaches. PPI defines IC contracts over the finite horizon
r0, T s, informally motivates the infinite-horizon model as a limit of the finite-horizon
model as T Ñ 8, and then directly defines the principal’s optimization problem, solves
for optimal contracts, and verifies that they are “IC” in the infinite-horizon model without
defining the agent’s feasible set F Ñ M of misreporting strategies in that context orchecking transversality conditions on the agent’s value function. 23 We describe these
22Approach (i) obviates the need to distinguish between LAC and GAC, but can be difficult to interpretbecause the limit contract may be neither IC nor optimal in the appropriately defined infinite-horizon
model (see, e.g., Prat and Jovanovic (2014, Section 4)). Approach (ii) is typically carried out by imposing
GAC (as in Sannikov (2014); DeMarzo and Sannikov (2016); Chen (2021)), which might be viewed as anoverly stringent assumption.
23PPI describes the infinite-horizon setting as follows (p. 1248): “I now turn to the principal’s problem
of optimal contract design over an infinite horizon. Formally, I take limits as T Ñ 8 in the analysis

16

omissions and their implications below.
Definition of Agent’s Strategy Space. As noted in Section 2.2, PPI’s formulationof the finite-horizon model explicitly assumes that the agent is restricted to strategies
inducing AC changes-of-measure over the finite horizon. In the infinite-horizon context,
PPI never specifies analogous restrictions, such as LAC or GAC. It can be shown that
GAC is not a viable assumption in conjunction with IML because, in the generic caseof transient shocks ( ° 0), truthtelling is the only feasible strategy satisfying bothconditions. 24 We therefore explore the consequences of LAC combined with IML, which
seems to provide the infinite-horizon version of PPI’s model that is most faithful to its
finite-horizon version, while still providing a fully formulated contracting problem. For
brevity, we henceforth refer to this combination of restrictions as “PPI’s assumptions.”
Transversality in Agent’s Problem. PPI’s main analysis of infinite-horizon IC (pp.
1271–72 in §A.3.2) pertains to Contract PPI under transient shocks ( ° 0). Therein,
PPI formulates the agent’s reporting problem as a stochastic control problem in which
t is a control variable and the current promised utility qt and misreport mt serve as
state variables, so that the agent’s value function at time t can be written as a function ofpqt , mt q. 25 PPI first conjectures that the agent’s value function takes the form
[4.1]

$
´
¯
⇢`
&q expp✓mq ¨⇢` `✓ mV W pq, mq :“%´8

for m P pM , 0s
for m § M ,[of finite-horizon IC contracts] above. Thus we no longer have the terminal conditions for the co-states[promised utility qT and marginal promised utility pT ] in (11) and (12) [on p. 1244 of PPI]; instead wehave the transversality conditions limT Ñ8 e´⇢T qT “ 0 and limT Ñ8 e´⇢T pT “ 0.” These transversality(or terminal) conditions should be understood to hold only under truthful reporting (i.e., P˚ -a.s. but notnecessarily Pm -a.s. for other m P F ) for two reasons. First, the terminal conditions of the finite-horizon
model analyzed in earlier sections of PPI apply only under truthful reporting. Second, more generally,
the first-order approach—which PPI derives via the Maximum Principle and other papers derive via the
Envelope Theorem (e.g., Pavan, Segal, and Toikka 2014)—only delivers necessary conditions for IC that
hold “on path” under truthful reporting. Thus, PPI’s transversality conditions make no reference to theagent’s feasible set and are not sufficient conditions for IC. To verify that a contract is IC, one should
instead use a transversality condition that holds under all feasible strategies to ensure that the agent’s
value function is sufficiently “continuous at infinity” that he cannot benefit from infinite-length deviations(cf. the proof of Theorem 3 in Pavan, Segal, and Toikka 2014).
24See Fact 3(i) in Appendix C, which implies that all contracts are M§GAC -IC. Meanwhile, the set M§GAC
GAC and MGAC contain many nontrivial
does contain nontrivial strategies when “ 0, and the sets M´strategies for all values of • 0. Thus, GAC is problematic only in conjunction with IML and ° 0.
25PPI also provides a separate argument based on an informal infinite-horizon adaptation of PPI’s
Theorem 4.1. That argument is also incorrect for reasons similar to those described here (see Appendix D).

17where M :“ ´p⇢ ` q{p✓ q. 26 This is the agent’s lifetime utility from always setting“ 0 irrespective of the current misreport m. PPI then shows that V W satisfies a suitable
HJB equation (p. 1272) and concludes from this that (a) V W is the agent’s value function
given the feasible set F “ M§LAC and (b) Contract PPI is M§LAC -IC.
Conclusions (a) and (b) are both unwarranted because the agent’s HJB equation
may have multiple solutions. Standard “verification theorems” in stochastic control
require verifying that V W satisfies the transversality condition
[TVC]

“
‰
lim Em e´⇢t V W pqt , mt q “ 0 for all m P F

tÑ8

before concluding that it is the agent’s value function, rather than some other solution to
the HJB equation. 27 PPI does not carry out this verification step. In fact, it follows from
[4.1] that V W violates [TVC] unless the feasible set F prevents the agent’s misreports“‰
mt from approaching M too quickly. For instance, limtÑ8 Em e´⇢t V W pqt , mt q “ ´8under any misreporting strategy that crosses M (and thereafter stays below) in finite
time. As there are many such strategies in M§LAC , one cannot draw either conclusion (a)or (b) under PPI’s assumptions using the standard guess-and-verify approach. 28
Consequences of Omissions. Unfortunately, conclusions (a) and (b) above are alsoboth incorrect. Formally, we show that the agent’s true value function is identically zerounder PPI’s assumptions, meaning that the agent can achieve near-infinite consumption
by deviating from truthtelling. Moreover, Contract PPI is not IC if the agent’s misreports≥Tare permitted to violate NP-m with rate ⇢ (i.e., if the cumulative misreports 0 m⌧ d⌧ candiverge to ´8 at an exponential rate faster than ⇢).
Observation 2. If ° 0, then Contract PPI satisfies the following properties:(i) The agent’s value function under feasible set F “ M§LAC is identically zero.
“‰(ii) It is not M§LAC X Mr -IC for any r ° ⇢.

Proof. Given any " ° 0 and  ° 0, define the “Ponzi scheme” misreporting strategy
26PPI does not mention the constant M or specify that V W pq, mq “ ´8 when m § M , apparently

intending for V W pq, mq to be defined as in the first line of [4.1] for all m § 0. However, the expression in
the first line of [4.1] is strictly positive whenever m † M , making it impossible for V W so-defined to be
the agent’s lifetime utility under any strategy (as the agent’s CARA utility function is strictly negative).
27For instance, see Pham (2009, Theorem 3.5.3). The importance of verifying transversality conditionsin problems with unbounded returns is also familiar from discrete-time dynamic programming (e.g.,
Stokey, Lucas, and Prescott 1989). The condition [TVC] differs from PPI’s transversality condition
limT Ñ8 e´⇢T qT “ 0 P˚ -a.s. (recall Footnote 23) in two ways. First, [TVC] concerns the conjectured
value function process V W rather than the promised utility process q , which do not coincide off-path.
Second, [TVC] concerns all feasible strategies, rather than just truthful reporting.
28The condition [TVC] is sufficient but not necessary for V W to be the true value function. By adapting
arguments from Appendix I, it can be shown that V W is, in fact, the agent’s true value function under IML
and the no Ponzi constraint [NP-m] with r “ ⇢, which allows for many strategies that violate [TVC].

18

mp",q as follows:
p",q

mt

:“

#
´t{"

´ept´"qfor t P r0, "s,for t ° ".

Under this strategy, the agent under-reports forever at an exponentially growing rate of (after initializing his misreport at time t “ " to m" “ ´1). This strategy is plainly in M§rfor all r °  and, being deterministic and bounded on each finite time horizon r0, T s,
is also in MLAC (cf. Appendix C). Next, notice that [3.4] allows us to write the agent’s
actual consumption process cm
t ” ct ´ mt under Contract PPI and any strategy m P M asm˚
cm`t “ ct`⇢ˆ ªt
˙⇢ m⌧ d⌧ ´ mt
0“: ⇠tmwhere cm is the agent’s consumption process under truthful reporting and the ⇠ m process
is proportional to the agent’s “extra consumption” from misreporting. For the strategy
mp",q , a simple calculation yields
˚mp",q

⇠t

“

# `

˘
1 ´ ⇢2 t`˘`˘ept´"q 1 ´ ⇢ ` ⇢ 1 ´ 2"t"

for t P r0, "q,for t P r", 8q.

It is easy to see that ⇠tm° 0 for all t ° 0 whenever  ° ⇢ and " P p0, "p⇢, qq, where
we let "p⇢, q :“ mint2{⇢, 2{u.
We may now prove Observation 2. Let , ⇢ ° 0 and r ° ⇢ be given. For any
 P p⇢, rq and " P p0, "p⇢, qq, it follows from the above that the agent derives strictly
greater consumption at all times t ° 0 from the strategy mp",q than from truthtelling.
This establishes part (ii). As for part (i), consider the sequence of strategies mp1{n,nq , sothat n :“ n Ñ `8 and "n :“ 1{n satisfies "n † "p⇢, n q for all n ° ⇢{2. By construction,p1{n,nq⇠tmÑ `8 for each fixed t ° 0. This implies that the agent’s actual consumption
mp1{n,nqctÑ `8 for each fixed t ° 0. It is then easy to see that the agent’s lifetime utility
under mp1{n,nq converges to its upper bound of zero as n Ñ 8. As each such strategy isin M§LAC , this establishes part (i) and thereby completes the proof.
p",q

Our Approach & the Necessity of Tail Restrictions. Observation 2(ii) implies thatthe no Ponzi constraint [NP-m] with some r § ⇢ is a (nearly) necessary condition forContract PPI to be IC. More generally, our analysis suggests that the no Ponzi constraint
is important for PPI’s model to be well-behaved, and we conjecture that it may benecessary for PPI’s model to admit IC contracts with nontrivial risk-sharing. 29 For these
29For instance, we show in Appendix H that without this constraint, Contract PPI cannot be implemented
as a direct mechanism but can be indirectly implemented in an alternative model where the agent has

19reasons, we henceforth focus on F -IC contracts for feasible sets F satisfying [NP-m].
4.3. Strict Suboptimality of Contract PPI
PPI states (p. 1254 in §6) that Contract PPI is optimal in the hidden endowment model.
This statement is problematic for two reasons. First, Observation 2 demonstrates thatContract PPI is not IC under PPI’s assumptions. Even if this could be addressed by
imposing sufficiently tight tail restrictions on the agent’s strategy space, it is a priori
unclear whether Contract PPI would be IC if IML were relaxed to NHB (or if sign
restrictions were dropped altogether). Second, and more importantly, we show that—independently of these issues—Contract PPI is strictly suboptimal whenever the agent’s
endowment process has non-zero mean reversion. (We describe PPI’s derivation ofContract PPI and the source of this discrepancy in Appendix A.) Formally:
Observation 3. If ° 0, then Contract PPI satisfies the following properties: 30(i) It is M⇢ -IC.
(ii) Given any r P p0, ⇢q, it is Mr -IC but strictly suboptimal among Mr -IC contracts.
Per Fact 1, the same is true if either NHB or IML is also imposed.
Proof. This follows from the analysis in Sections 5 and 6 below. Specifically, part (i)is an immediate corollary of Theorem 2 in Section 6, while part (ii) is an immediate
corollary of that result in conjunction with Theorem 1 in Section 5.
Observation 3 has three implications. First, in conjunction with Observation 2, it
demonstrates that restricting the agent’s strategy space to M⇢ is a necessary and sufficientcondition for Contract PPI to be IC. Second, it establishes that Contract PPI is strictly
suboptimal under the slightly stronger assumption that the agent’s strategy space is
Mr for some r P p0, ⇢q. Third, a corollary of these observations and Fact 1 is that the
incentive compatibility and strict suboptimality of Contract PPI are both independent of
the sign restrictions that one imposes on the agent’s feasible set.
access to hidden savings (see Section 5.4), which is unreasonable because the agent has access to moredeviations in the latter model. Subsequent to our working paper, Acciaio, Crowell, and CvitaniÊ (2022)show that, without any tail restrictions on the agent’s misreports, deterministic contracts—which do
not condition on the agent’s reports, and hence provide no insurance—are optimal within a class of“linear contracts” that includes the SI Contracts that we study in Sections 5 and 6. (Acciaio, Crowell, andCvitaniÊ (2022) also verify that, under suitable tail restrictions, the optimal SI Contract that we identify in
Theorem 1 is optimal among all linear contracts.) We view this as evidence that, for technical reasons, tailrestrictions are needed for PPI’s model to deliver a non-degenerate contracting problem.
30We state Observation 3 in this two-part manner to emphasize the weakest technical conditions underwhich we are able to (separately) establish the incentive compatibility and suboptimality of Contract PPI.
The stronger, and standard, GAC assumption is sufficient for both parts of Observation 3.

20

Our Approach & Self-Insurance. The formal analysis underlying Observation 3 issomewhat involved, but the basic idea is simple. Under truthful reporting, the consumption process induced by Contract PPI satisfies the agent’s Euler equation: plugging [3.4]
into the agent’s marginal utility u1 pcq “ ´✓upcq yields
u1 pct q “ Et ru1 pc⌧ qs for all ⌧ • t,

[4.2]

so the agent’s marginal utility is a martingale under Contract PPI. This familiar equation
represents the agent’s optimal consumption-saving behavior in a setting where (a) there
is no principal and (b) the agent self-insures by investing in a risk-free bond with interest
rate r “ ⇢. The prior literature shows that optimal contracts in discrete-time analogues of
PPI’s model typically do not coincide with solutions to self-insurance problems (Thomas
and Worrall 1990), except in an alternative model in which the agent also has access to
hidden savings (Allen 1985; Cole and Kocherlakota 2001). The crux of our approach is
to argue that this classic finding remains true in PPI’s model.
�. A Self-Insurance Approach

Section 5.1 presents a standard self-insurance problem for the agent. Section 5.2 uses thisproblem to construct an indirect implementation for our main class of “self-insurance
contracts.” Section 5.3 shows that this class contains Contract PPI and characterizes the
optimal self-insurance contract. Section 5.4 summarizes additional results.
5.1. Agent’s Self-Insurance Problem
Consider the classic self-insurance problem faced by the agent with CARA utility
upcq “ ´e´✓c when there is no principal to provide insurance. In this problem, the agent
receives only (i) his endowment stream b and (ii) some initial asset holdings A0 P R,
and must self-insure by borrowing and saving in a risk-free bond market at the given
interest rate r ° 0. Formally, the agent solves 31
[5.1]

SI

V pA0 , b0 q :“supĉP ApA0 ,b0 q

E0

„ª 8

e

´⇢t

0

upĉt q dt⇢

where ApA0 , b0 q is the set of pA0 , b0 )-feasible consumption strategies ĉ “ pĉt qt•0 , which
consists of all consumption processes that are b-adapted and induce an asset process` ˘Aĉ “ Aĉt t•0 that solves[5.2]

dAĉt “ prAĉt ` bt ´ ĉt q dt

31Recall that P denotes the probability measure over paths of b induced by the law of motion [2.1], andlet Et denote the associated conditional expectation operators.

21

subject to the initial condition Aĉ0 “ A0 and the no Ponzi condition
lim e´rt Aĉt • 0[NP-A]

tÑ8

P-a.s.

A consumption strategy ĉ is optimal in the agent’s self-insurance problem if it attains
the supremum in [5.1], where V SI pA0 , b0 q is the agent’s self-insurance value function.
Lemma 5.1. The consumption strategy ĉ˚ defined by[5.3]ĉ˚t :“ ĈpA0 , b0 q `

ˆ

r´⇢`

f pr; q2 {2
✓

2

˙

t`

f pr; qWt✓

is optimal in the agent’s self-insurance problem, 32 where f pr; q :“ r✓{pr ` q andrb ´ Āpr; qr`r ´ ⇢ ` 2 f pr; q2 {2µĀpr; q :“
´
.
r✓
r`

[5.4]

ĈpA, bq :“ rA `[5.5]

The proof of Lemma 5.1 is in Appendix E. The consumption strategy in [5.3]–[5.5]is the continuous-time limit of the discrete-time self-insurance solutions in Caballero(1990) and Wang (2003), and was previously derived in continuous time by Wang (2004,
2006). Our derivation involves slightly different arguments than in this prior work,
allowing us to dispense with some technical conditions imposed therein. 33
Lemma 5.1 implies that the agent’s optimal consumption strategy can be expressed˚recursively as ĉ˚t ” Ĉ pA˚t , bt q, where A˚t :“ Aĉt is the induced asset process. Specifically,at each time t, the agent consumes a multiple r of his permanent income[5.6]

A˚t ` Et

„ª 8
t

e

´rp⌧ ´tq

b⌧ d⌧

⇢

1
“ A˚t `

„

r
bt `
r r`

µ
`r

⇢

adjusted by subtraction of a constant term. This facilitates a natural interpretation of
the agent’s risk exposure, viz., the sensitivity f pr; q{✓ “ r{pr ` q of consumption to
endowment shocks in [5.3]. Observe that the derivative of the agent’s permanent income[5.6] with respect to his current endowment bt is 1{pr ` q. Because wealth effects areabsent under CARA utility, the agent optimally responds to a marginal increase in his
32Furthermore, this strategy is uniquely optimal (P-a.e.) because the set of feasible consumption
strategies is convex (see [E.1] in Appendix E) and the agent’s objective function is strictly concave.
33Wang (2004, 2006) imposes a technical integrability condition on the space of admissible asset
processes, as well as a transversality condition on a function that is conjectured to be the agent’s valuefunction (and verified to be so under these assumptions). Our argument instead works directly with theagent’s true value function and only requires that the asset process satisfies [5.2] and [NP-A]. This allows
us to construct an exact mapping between the agent’s consumption strategies in the indirect “self-insurance
contracts” of this section and his reporting strategies in their direct revelation counterparts (see Section 6).

22

endowment by permanently shifting up his future consumption by a constant amount ofr{pr ` q at all future dates. This increases the present value of consumption by 1{pr ` q,
exactly matching the increase in permanent income.
It will be useful in what follows to note the agent’s risk exposure is strictly increasing in r when ° 0 but is constant in r when “ 0. Intuitively, as the coefficient
f pr; q depends only on the ratio {r, that ratio is the endowment’s “interest-adjusted
rate of mean reversion.” For a fixed ° 0, the adjusted rate {r is decreasing in r, sothat increasing r increases the “effective persistence” of endowment shocks. However,when “ 0, endowment shocks remain perfectly persistent regardless of r.
Lemma 5.1 has several further implications that will prove useful (details are in
Appendix E). First, the agent’s optimal strategy satisfies the familiar Euler equation
[5.7]

“
‰
epr´⇢qt u1 pĉ˚t q “ Et epr´⇢q⌧ u1 pĉ˚⌧ q

for all ⌧ ° t,

which specifies that the agent’s discounted marginal utility is a martingale (and reduces
to [4.2] when r “ ⇢). Second, the agent’s continuation value process V “ pVt qt•0 definedby Vt :“ V SI pA˚t , bt q satisfies
[5.8]
[5.9]

upĉ˚t q
“ Vt
r

„ ˆ
“ V0 exp ´ r ´ ⇢ `2f pr; q22˙⇢t ´ f pr; q Wt .

Display [5.8] implies that the agent’s self-insurance solution induces a constant utility
delivery rate of r, 34 while [5.9] implies that the agent’s continuation value process Vt is ageometric Brownian motion and the discounted value process epr´⇢qt Vt is a martingale.
Finally, plugging the optimal strategy [5.3] into the agent’s objective [5.1] lets us express
the agent’s self-insurance value function as[5.10]SI

V pA0 , b0 q “ V̂
`

˘

SI

„

ˆ
exp ´✓r A0 `

b0
r`

˙⇢

where V̂ SI :“ ´ 1r exp ✓Āpr; q is a constant.
5.2. Self-Insurance Contracts

We now introduce a class of contracts defined by an indirect implementation in which
the principal acts as the agent’s “bank” by (i) giving an initial lump-sum (asset) transfer
to the agent and then (ii) allowing the agent to self-insure at interest rate r ° 0, which
34This property arises because, under CARA utility, the Euler equation [5.7] implies that discounted
flow utility epr´⇢qt upĉ˚t q is itself a martingale.

23

is determined by a constant marginal tax (if r † ⇢) or subsidy (if r ° ⇢) imposed on
the agent’s savings. (Recall from Section 2.1 that ⇢ can be interpreted as the market
rate at which the principal finances the contract.) In this implementation, the agent isonly able to borrow and save via his “account” with the principal—not in the ambient
market—so that the principal can observe, and hence tax, the agent’s asset holdings. The
principal does not observe the agent’s endowment or actual consumption. There is nocommunication: the agent does not submit endowment reports.
Definition 5.2. The Self-Insurance Contract (SI Contract) pb0 , q0 , rq is the indirect mechanism consisting of the following steps:(i) The principal gives the agent initial assets[5.11]

A0 pb0 , q0 , rq :“cpq0 , rq
b0
´
r
r`

`

Āpr; qr

where cpq, rq :“ ´ logp´rqq{✓ and Āpr; q is defined in [5.5].
(ii) The principal, acting as the agent’s bank, allows the agent to solve his self-insurance
problem at the rate r (as defined in Section 5.1) by imposing a constant marginal tax(or subsidy) ⌧ prq :“ 1 ´ r{⇢ on the agent’s capital gains ⇢At . The principal thereby
collects tax revenue ⌧ prq⇢At at each instant.
“≥8‰
The principal’s expected lifetime cost is ⇧pb0 , q0 , rq :“ A0 ´ E0 0 e´⇢t ⌧ prq⇢At dt .

Clearly, the agent’s best-response when faced with the SI Contract pb0 , q0 , rq is tofollow the consumption-saving strategy described in Lemma 5.1 for the induced selfinsurance problem with initial assets A0 pb0 , q0 , rq and rate r. It is then easy to verify from`˘[5.8] that the agent’s initial lifetime utility satisfies V0 “ 1r u ĈpA0 pb0 , q0 , rq, b0 q “ q0 , sothat the contract does, in fact, deliver the requisite promised utility. Plugging the agent’s
optimal strategy into the principal’s cost function, we find that the principal’s cost of the
SI Contract pb0 , q0 , rq admits a simple closed-form expression.
Lemma 5.3. The principal’s cost ⇧pb0 , q0 , rq of the SI Contract pb0 , q0 , rq satisfies
2 2logp⇢{rq r ´ ⇢✓r⇧pb0 , q0 , rq “ J pb0 , q0 q `
`
`
2
2
⇢✓
✓⇢
2⇢ pr ` q2
˚

[5.12]

“ E0

„ª 8
0

e

´⇢t

pĉ˚t ´ bt q dt⇢“

2 f pr;`˘
q2 { 2✓⇢2where J ˚ pq0 , b0 q is the principal’s first-best cost function and ĉ˚ is defined in [5.3]. 35
35That is, J ˚ pb0 , q0 q is the principal’s cost of providing full insurance in the full-information problem

where the agent’s endowment is observable and contractable. It is easy to show, as in PPI (p. 1252), thatJ ˚ pb, qq “ rp⇢ ` qc̄pq, ⇢q ´ µ ´ ⇢bs{⇢p⇢ ` q, where c̄pq, ⇢q “ ´ logp´⇢qq{✓ as defined above in [5.11].

24

The proof of Lemma 5.3 is in Appendix F. Display [5.12] states that the principal’s
cost of the SI Contract pb0 , q0 , rq, which is defined in terms of the initial wealth transfer
and subsequent tax revenue, coincides with the expected resource cost of the contract,
as defined in [2.2] for truthful direct revelation contracts. 36
5.3. Contract PPI and the Optimal SI Contract
We now establish the main result of this section: Contract PPI can be indirectly implemented as an SI Contract with zero taxes (r “ ⇢), whereas the optimal SI Contract
features strictly positive taxes (r † ⇢) whenever endowment shocks are transient ( ° 0).
Consequently, Contract PPI is generically suboptimal within the class of SI Contracts.
Implementing Contract PPI as an SI Contract. To begin, we observe that the agent’s
consumption process and the principal’s costs are identical under (i) Contract PPI if
the agent reports truthfully and (ii) the corresponding SI Contract with rate r “ ⇢ if theagent follows his optimal consumption-saving strategy.
To verify this, compare the agent’s promised utility process q and recommended
consumption process c under Contract PPI (recall [3.3] and [3.4]) to his value function
process V and optimal consumption strategy ĉ in the self-insurance problem with interest
rate r “ ⇢ (recall [5.3] and [5.9]). If the agent follows the truthful reporting strategy m˚ in˚Contract PPI, then these processes are identical: q “ V and c “ cm “ ĉ. By Lemma 5.3,
the indirect implementation costs the principal[5.13]⇧pb0 , q0 , ⇢q “ J ˚ pb0 , q0 q `2✓,
2p⇢ ` q2

which coincides with PPI’s expression (p. 1253) for the principal’s lifetime cost (asdefined in [2.2]) under Contract PPI.
In other words, Contract PPI (under truthful reporting) is outcome-equivalent toletting the agent self-insure at the ambient market rate ⇢. This equivalence allows us
to reinterpret the two main economic findings that PPI derives from Contract PPI interms of known results from the self-insurance literature. First, as stated in Fact 2(i),PPI finds that Contract PPI leads to long-run bliss: ct Ñ `8 and qt Ñ 0 almost surely.
Correspondingly, in the context of self-insurance, it is known quite generally that when
the interest rate satisfies r • ⇢, the agent’s desire to accumulate precautionary savings
leads him to both save and consume without bound: A˚t , ĉ˚t Ñ `8 P-a.s. (Sotomayor(1984); Chamberlain and Wilson (2000); Ljunqvist and Sargent (2000, Ch. 17)). Second,
PPI (pp. 1254–55) finds that, under Contract PPI, the quality of risk-sharing degrades as
36Furthermore, any SI Contract can be implemented using only flow transfers (as in Section 2) byreplacing the lump-sum transfer of A0 with a deterministic transfer process ŝ satisfying dŝt :“ p↵ ´ ŝt q dt
for suitably chosen parameters ŝ0 , ↵ P R.

25

the agent’s endowment becomes more persistent (i.e., f p⇢; q is decreasing in ). This
comparative static is known to arise from the agent’s optimal consumption-smoothing
in the corresponding self-insurance problem (Caballero (1990); Wang (2003)).
The Optimal SI Contract. This discussion suggests a simple way to improve upon
Contract PPI. Namely, Lemma 5.3 and a short calculation reveal thatd⇧pb0 , q0 , rq|r“⇢ • 0, with strict inequality if and only if
dr

° 0.

Thus, whenever ° 0, some SI Contract pb0 , q0 , rq with r † ⇢ delivers the same lifetime
utility q0 to the agent at a strictly lower cost to the principal than [5.13]. In other words,
Contract PPI is generically improvable by taxing the agent’s savings.
Theorem 1. For any initial condition pb0 , q0 q, there exists an optimal SI Contract
pb0 , q0 , r˚ q, where r˚ is a minimizer of ⇧pb0 , q0 , ¨q from [5.12]. 37 It satisfies the following:(i) If ° 0, then r˚ † ⇢ and the optimal SI Contract has a strictly lower cost thanContract PPI.
(ii) If “ 0, then r˚ “ ⇢ and the optimal SI Contract implements Contract PPI.
Proof. Let pb0 , q0 q be given. It is easy to see that limrÑ0 ⇧pb0 , q0 , rq “ limrÑ8 ⇧pb0 , q0 , rq “
8 and that ⇧pb0 , q0 , ¨q is continuously differentiable on R`` . Consequently, there exists a
minimizer r˚ P R`` , and any such minimizer satisfies the necessary first-order conditiond⇧pb0 , q0 , rq|r“r˚ “ 0, which can be expressed as
dr[5.14]„⇢ˇdr ˇˇ´ logprq `
ˇ
dr
⇢ ˇ

r“r

d
`
dr
˚

° 0 if„ 2 2⇢ ˇ
f pr; q ˇˇˇ
ˇ2⇢
° 0, “ 0 if

“0

r“r˚

“ 0.

When ° 0, the second term in [5.14] is strictly positive, implying that the first term isstrictly negative. It follows that any minimizer satisfies r˚ † ⇢, delivering part (i). When
“ 0, the second term in [5.14] is identically zero and the objective ⇧pb0 , q0 , ¨q is strictly
convex. Thus, the unique minimizer is r˚ “ ⇢, delivering part (ii).
To get additional intuition for Theorem 1, observe that under an SI Contract withinterest rate r, the agent’s continuation utility process V from [5.9] evolves as[5.15]

dVt“ p⇢ ´ rqdt ´ f pr; q dWt .
VtIntuitively, [5.15] reflects two distortions away from first-best insurance provision: the“drift distortion” determined by the difference |⇢ ´ r| ° 0 and the “risk distortion”
37Going forward, we slightly abuse terminology by referring to the optimal SI Contract. The optimumis always unique when “ 0 and, when ° 0, is unique for generic specifications of model parameters.

26

determined by the risk exposure coefficient f pr; q ° 0. The optimal SI Contract resultsfrom the principal’s optimal trade-off between these two distortions. Note that whenr “ ⇢, as in Contract PPI, there is non-zero risk exposure but zero drift distortion,corresponding to PPI’s observation that the agent’s promised utility under Contract PPI
is a martingale (Fact 2(iii)). There are two cases to consider:(a) Transient endowment shocks ( ° 0): In this case, the risk exposure coefficient f pr; q
is strictly increasing in r. Therefore, marginally decreasing r from the value r “ ⇢has two effects: it creates a non-zero drift distortion while decreasing the agent’s
risk exposure. The principal’s first-order condition [5.14] equates the marginal cost of
the former effect (first term) with the marginal benefit of the second effect (secondterm), leading to an optimal rate of r˚ † ⇢. 38 This corresponds to imposing a strictly
positive tax on the agent’s savings, consistent with the literature’s finding that optimal
contracts in related insurance settings feature a “savings wedge” that relaxes theagent’s IC contraints (e.g., Golosov, Kocherlakota, and Tsyvinski (2003)).
(b) Permanent endowment shocks ( “ 0): In this case, the risk exposure coefficientf p¨; 0q ” ✓ is constant in r. Consequently, the principal cannot manipulate the agent’s
risk exposure by imposing a tax or subsidy on savings, as reflected by the fact thatthe second term in the principal’s first-order condition [5.14] vanishes. Moving raway from the value r “ ⇢ therefore only generates the cost of a non-zero drift
distortion, without creating any risk-reduction benefit. To eliminate this cost, theprincipal optimally sets r “ ⇢ and thereby implements Contract PPI.
In Section 6 below, we show that SI Contracts are IC when reformulated as directrevelation contracts. This analysis, combined with Theorem 1, completes the proof ofour main observation that Contract PPI is generically suboptimal (Observation 3).
5.4. Further Analysis
Properties of SI Contracts. In Appendix G, we establish some additional useful facts
about SI Contracts. We summarize them here:(i) The class of SI Contracts is characterized by the property, which we call Stationarity,that promised utility and marginal promised utility are proportional, viz., pt ” k0 qt
for some constant k0 ° 0. Per Fact 2(ii), Contract PPI plainly has this property. As
noted in Appendix A, the key incorrect step in PPI’s derivation of Contract PPI
involves arguing that restricting to Stationary Contracts is without loss of optimality.
We show that this conclusion is false: Theorem 1 is equivalent to the statement thatContract PPI is optimal within the class of Stationary Contracts if and only if “ 0,contradicting that Contract PPI is generally optimal among all IC contracts.
38It is never optimal to set r ° ⇢ because doing so creates a non-zero drift distortion while also
increasing the agent’s risk exposure, both of which are costly to the principal.

27

(ii) The class of SI Contracts coincides with the class of “state-consistent” renegotiationproof contracts introduced by Strulovici (2022). This equivalence provides a separate
foundation for our focus on SI Contracts based on the principal’s limited commitment.
It also shows that SI Contracts provide a simple indirect implementation for the fullclass of renegotiation-proof contracts in PPI’s model.
(iii) The long-run behavior of the optimal SI Contract depends starkly on the persistenceof the agent’s endowment: there exists a threshold ° 0 such that this contract resultsin immiseration when persistence is sufficiently low ( ° ) and results in bliss
when persistence is sufficiently high ( † ). This implies that the long-run bliss
property of Contract PPI is generally not robust to optimization within the class of
SI Contracts. This result may also be of independent interest, given the literature’s
emphasis on long-run properties of optimal contracts.
Hidden Savings. PPI adopts the standard assumption that the agent cannot save or
borrow outside of his relationship with the principal. Our SI Contracts rely on thisassumption: if instead the agent could covertly trade at the market rate ⇢, he could circumvent any tax imposed by the principal. Since Contract PPI is implemented with zerotaxes, this suggests that it may be uniquely robust to such manipulation. We formalize
this idea in Appendix H, where we consider an alternative version of PPI’s model withhidden savings. In this setting, Theorem 4 shows, in essence, that every IC contract
is outcome-equivalent to Contract PPI, which is therefore the optimal contract. This
parallels Allen’s (1985) and Cole and Kocherlakota’s (2001) classic discrete-time characterizations of optimal hidden savings contracts and provides a viable foundation forContract PPI, albeit one that is rather different than suggested in PPI.
�. Incentive Compatibility of SI Contracts

In Section 5, we characterized the agent’s optimal consumption strategy in an indirect
implementation. We now reformulate SI Contracts as direct mechanisms and showthat they are IC, i.e., truthtelling is the agent’s optimal reporting strategy. Our analysis
addresses the two issues in PPI related to IC (Observations 1 and 2) and introduces some
new techniques that may be applicable in other continuous-time models.
6.1. Extended Reporting Problem
We will see below (Section 6.4) that, if the agent’s recent reports are not truthful(lim"Œ0 mt´" ‰ 0), then he has an incentive to “instantly” revert to truthtelling (setmt “ 0). This requires him to submit a discontinuous “jump report.” While PPI’s formulation allows the agent to approximate such behavior (by sending | t | Ñ 8 for avanishing measure of times), discontinuous report paths are ruled out by construction.
Thus, for the purpose of analyzing IC, we find it convenient in this section to “extend”
28

PPI’s model formulation (from Section 2.1) in two ways: 39(i) We allow the agent’s feasible set F of misreporting strategies to include m R M.
(ii) We allow the principal’s contract, the y -adapted transfer process s, to respond in any
way we choose to y processes corresponding to m P F zM. To maintain consistency
with Section 2.1, we still require y processes corresponding to m P M to induce
continuous sample paths of s.
This approach is justified by an elementary variant of Fact 1: if a contract as definedin (ii) above is F -IC for a given feasible set F of b-adapted misreporting strategies(not necessarily satisfying F Ñ M), then the contract is also rF X Ms-IC. If the agent
does not have a profitable deviation from truthtelling within the “extended” feasible set
F , then he cannot have a profitable deviation within the “actual” (smaller) feasible set
F X M. Therefore, if the goal is to verify that a contract is rF X Ms-IC, then we are free
to “extend” the contract arbitrarily to paths of y that correspond to “infeasible” paths ofm P F zM, and then verify that the resulting “extended” contract is F -IC. 40 , 41
6.2. Direct Revelation SI Contracts
We can recast any SI Contract as a direct-revelation contract as follows:
Definition 6.1. The Direct-Revelation Self-Insurance Contract (DR-SIC) pb0 , q0 , rq isthe direct revelation mechanism in which:(i) The principal keeps track of the y -adapted virtual asset process Av defined by[6.1]

Avt :“ A0 pb0 , q0 , rq ` Āpr;

qt `

r`

ªt

y⌧ d⌧

0

where A0 pb0 , q0 , rq is defined in [5.11] and Āpr; q is defined in [5.5].
(ii) The agent’s recommended consumption is the y -adapted process c defined by[6.2]

ct :“ Ĉ pAvt , yt q “ rAvt `ryt ´ Āpr; q`r

39This approach builds on ideas from Strulovici (2022). While this approach is convenient, it is notnecessary: Section 7.3 and Appendix I describe how to analyze IC under PPI’s assumption that F Ñ M.
40One possible “extension,” described in Footnote 16, involves “shooting” the agent if his strategym P F zM generates an “immediately detectable” deviation, such as a discontinuous sample path of reportsŷ R Cr0, 8q, with positive probability. While this extension is without loss of generality, it is “trivial” inthe sense that it effectively prevents the agent from using m R M. In Section 6.2–6.4 below, we find it
more useful to consider an extension that “compactifies” the agent’s reporting problem by considering
a “closure” of the original contract that treats jump reports as the limit of “approximate jump” reports(whereby | ⌧ | Ñ 8 for a vanishing measure of times).
41When the feasible set F permits m with discontinuous sample paths, we can no longer view the agent
as choosing a measure Pm on the space C pr0, 8qq of continuous paths (cf. Footnote 11). This has noadverse consequence for our analysis in Section 6.2–6.4, which does not use any change of measure.

29

where Ĉ pA, yq is defined in [5.4].

Put simply, in a DR-SIC the principal “saves” on the agent’s behalf and recommends
that he consume as would be optimal in his self-insurance problem, assuming that theagent is truthful so that y ” b. We emphasize that virtual assets are not a “physical”
object, but simply a state variable that the principal uses to track what the agent would
have done in his self-insurance problem. 42
Observe that every DR-SIC: (a) is well-defined under any report process y thatinduces a well-defined virtual asset process Av in [6.1], and (b) specifies continuous transfers whenever y is continuous. Denote the corresponding set of extended misreporting
strategies by
[6.3]

"

Mext :“ m : m is b-adapted and

ªt
0

*

|m⌧ |d⌧ † 8 P-a.s. @t • 0 .

Following our treatment of tail restrictions from Section 2.2, we also define, for everyrr ° 0, the subset of extended strategies Mext
à Mext by
[6.4]

r
Mext
:“ tm P Mext : [NP-m] holds for rate ru .

We will study the “extended” reporting problem in which the agent’s feasible set isrr
F “ Mext(for a suitable value of r ° 0). Clearly, Mext
â Mr because [6.3]–[6.4] allow
for strategies with discontinuous sample paths. Notably, any DR-SIC responds to a“jump report” at time t (whereby mt ´ lim"Œ0 mt´" “ M ‰ 0) as if it were the limit of“approximate jump reports” at time t (whereby ⌧ “ M {" for ⌧ P rt, t ` "q and " Œ 0). 43
Using this notation, we can state the main result of this section as follows:r
Theorem 2. For any given r ° 0, every DR-SIC pb0 , q0 , rq is Mext
-IC. Thus, such conr
tracts are also F -IC for any smaller strategy space F Ñ Mext .

The assumptions in Theorem 2 cannot be substantively relaxed: by mimicking the“1‰
proof of Observation 2, it is easy to show that no DR-SIC with rate r is M§LAC X Mr -IC
for any r1 ° r. We conclude that sufficiently tight no Ponzi constraints are essential for
DR-SICs to be IC (while sign and absolute continuity restrictions are not).
We provide two independently instructive proofs of Theorem 2 below. Section 6.3
presents an indirect “revelation principle” argument in which we characterize the mapping between reporting strategies in a DR-SIC and consumption-saving strategies in
42As defined above, DR-SICs are recursive in the state variables pAvt , yt q, viz., the y -adapted transfer

process s can be written as st “ ĈpAvt , yt q ´ yt . By defining promised utility as qt :“ V SI pAvt , yt q (where
V SI is defined in [5.10]), we can equivalently write any DR-SIC recursively in the state variables pqt , yt q,
as in PPI’s treatment. In this alternative formulation, it is easy to see that a DR-SIC with interest rate ⇢coincides with Contract PPI (see Definition 3.1) under any reporting strategy.
43See Lemma B.1 and Remark B.2 in Appendix B.

30

the corresponding indirect SI Contract. This approach clarifies that the no Ponzi condition [NP-m] on the agent’s misreports in the direct mechanism is equivalent to theno Ponzi condition [NP-A] on the agent’s assets in the indirect mechanism. Section 6.4
then presents an alternative argument in which we directly analyze the agent’s reporting
incentives in a DR-SIC using stochastic control. This approach elucidates the agent’s
reporting incentives at non-truthful histories.
6.3. Indirect Proof: Revelation Principle
We establish Theorem 2 in two steps. First, we characterize the agent’s actual consumption process cm in a DR-SIC under an arbitrary misreporting strategy m P Mext
(Lemma 6.2). Second, we show that an actual consumption process cm can be inducedr
in a DR-SIC with interest rate r by some strategy m P Mext
if and only if cm is afeasible consumption strategy for the agent in the corresponding indirect SI Contract(Lemma 6.3). Theorem 2 is then immediate: the absence of profitable deviations in
the indirect implementation implies the absence of profitable deviations in the direct
mechanism.
Step 1: Induced Consumption Processes. Let ĉ˚ denote the agent’s optimal consumption strategy in the SI Contract pb0 , q0 , rq and let A˚ denote the corresponding asset
process, so that ĉ˚t “ ĈpA˚t , bt q as in Lemma 5.1. The following lemma characterizes how
misreports in the direct mechanism correspond to consumption choices in the agent’s
self-insurance problem.
Lemma 6.2. Given any misreporting strategy m P Mext , the agent’s actual consumption
cm
t :“ ct ´ mt in the DR-SIC pb0 , q0 , rq satisfies
[6.5]
[6.6]

v
cm
mt
t “ ĈpAt , bt q ´
r
`
ˆ ªt
˙
˚
“ ĉt `
r m⌧ d⌧ ´ mt .
r`0

Proof. Display [6.5] follows from recalling the identity cm
t ” ct ´ mt and substituting
the identity yt ” bt ` mt into the expression for ct in [6.2]. To obtain [6.6], observe from≥t[6.1] that Avt ” A˚t ` r` 0 y⌧ d⌧ and substitute this identity into [6.5].

Display [6.5] implies that under-reporting by setting mt † 0 in the DR-SIC corresponds to over-consuming by ´ r` mt ° 0 in the indirect SI Contract, holding fixed theagent’s assets at Avt . At the same time, because virtual assets evolve as
[6.7]

dAvt
dA˚t
“
`
mt ,
dt
dt
r`

31

an under-report of mt † 0 in the DR-SIC corresponds to under-saving by r` mt † 0 in
the indirect SI Contract. Display [6.6] describes how these virtual savings distortions
accumulate over time, so that the agent’s actual consumption cm
t in the DR-SIC may be˚either higher or lower than his optimal consumption ĉt in the indirect SI Contract.
Step 2: Outcome-Equivalence of Feasible Sets. The following lemma relates thesets of feasible consumption processes in the direct and indirect implementations.
Lemma 6.3. For any b-adapted consumption process ĉ, the following are equivalent:(i) It is feasible in the SI Contract pb0 , q0 , rq.
r(ii) It satisfies ĉ “ cm for some misreporting strategy m P Mext
in the corresponding
DR-SIC pb0 , q0 , rq.
Proof. That (ii) implies (i) is immediate from the definitions. To see that (i) implies (ii),suppose that ĉ is feasible in the SI Contract pb0 , q0 , rq and let Aĉ be the corresponding
asset process (i.e., solution
to [5.2]ı–[NP-A]). Motivated by [6.5], define the misreporting
”
r`
ĉ
ĉ
strategy mt :“ĈpAt , bt q ´ ĉt , which is in Mext by construction. 44 We claim thatmĉ induces the virtual asset process Av “ Aĉ and actual consumption process cm “ ĉ inthe DR-SIC pb0 , q0 , rq. Substitute mĉ into [6.1], expand the definition of Ĉ from [5.4], and˘≥t `recall that Aĉ (being a solution to [5.2]) satisfies Aĉt “ A0 pb0 , q0 , rq` 0 rAĉ⌧ ` b⌧ ´ ĉ⌧ d⌧ .
After simplification, this yields Avt ” Aĉt . Since Aĉ satisfies [NP-A], it follows that mĉ
rsatisfies [NP-m], and thus mĉ P Mext
. Finally, substituting Avt ” Aĉt and the definition ofĉ
mĉ into [6.5] (from Lemma 6.2) yields cm” ĉt , completing the proof.
tĉ

The key observation underlying Lemma 6.3 is that, in a DR-SIC, the no Ponzi
condition [NP-m] on misreports is equivalent to the no Ponzi condition [NP-A] on his
virtual asset process Av . Formally, given any misreporting strategy m P Mext , the limits≥T
limT Ñ8 e´rT AvT and limT Ñ8 e´rT 0 mt dt are P-a.s. equal. 45 Since it is well-understoodthat [NP-A] is necessary for the agent’s self-insurance problem to be well-posed, thissuggests that [NP-m] is a natural constraint on misreporting strategies in DR-SICs.
Proof of Theorem 2. By definition, ĉ˚ is the agent’s optimal consumption process
in the indirect SI Contract pb0 , q0 , rq. Thus, Lemma 6.3 implies that any misreportingr
strategy m P Mext
for which cm “ ĉ˚ is optimal for the agent in the corresponding
44Clearly, mĉ is b-adapted. To verify that it is integrable, expand the definition of≥ Ĉ from [5.4],≥ regroup

terms, and use the facts that b solves [2.1] and Aĉ solves [5.2] to conclude that 0 |mĉ⌧ |d⌧ § 0 |rAĉ⌧ `≥t≥tb⌧ ´ ĉ⌧ |d⌧ ` 0 |Āpr; q|d⌧ ` r` 0 |b⌧ |d⌧ † 8 P-a.s.
t

t

≥T

45To see this, integrate the “virtual savings” equation [6.7] to obtain AvT “ A˚T ` p ` rq´1 0 mt dt,

where A˚ is the asset process in the corresponding SI Contract induced by the agent’s optimal consumptionsaving strategy therein (Lemma 5.1). We have limT Ñ8 e´rT A˚T “ 0 because the agent optimally “leaves
no money on the table” in his self-insurance problem (see Lemma E.3 in Appendix E).

32

DR-SIC pb0 , q0 , rq. By [6.6] in Lemma 6.2, the truthful strategy m˚ ” 0 is one suchr
strategy. It follows that the DR-SIC pb0 , q0 , rq is Mext
-IC.
6.4. Direct Proof: Dynamic Programming
We now establish Theorem 2 by directly analyzing the agent’s reporting incentives. 46
Proof of Theorem 2. Fix a DR-SIC with interest rate r. By [6.5] in Lemma 6.2, theagent’s actual consumption process cm is Markovian in the contemporaneous virtual
assets Avt , true endowment bt , and misreport mt . Thus, the agent’s reporting problemr(with strategy space Mext
) can be cast as one of stochastic control in which pAvt , bt q serveas state variables and mt is a control variable. Let V DR pAvt , bt q denote the agent’s valuefunction in this control problem. By standard arguments, it can be shown that 47
[6.8]

V

DR

pAvt , bt q “ V̂ DR exp„ˆ´✓r Avt `

bt
r`

˙⇢

for some constant V̂ DR † 0, and that V DR pAvt , bt q is a solution to the HJB equation
⇢V

DR

[6.9]

pAvt , bt q “ sup
mt PR

” ˆ
u ĈpAvt , bt q ´

mt
r`

˙

ˆ
˙
ı
pbt ` mt q
DRv
` Āpr; q `VA pAt , bt qr`

` pµ ´ bt qVbDR pAvt , bt q ` 12 2 VbbDR pAvt , bt q.

Routine analysis of [6.9] then yields a full characterization of the agent’s optimal strategy:
since the supremum in [6.9] is uniquely attained by setting mt “ 0, the agent’s uniqueoptimal strategy is to truthfully report his current endowment at every history. 48 Itrfollows that the given DR-SIC with interest rate r is Mext
-IC.
This proof implies that the agent truthfully reports his current type bt (by setting
46Strulovici (2022) uses a related argument to analyze IC for an equivalent class of renegotiation-proof
contracts (see Section 5.4). A notable technical difference is that Strulovici (2022) casts the agent’s
reporting problem as one of impulse-control, viewing m as a state variable (as in PPI’s treatment) andas an extended-real-valued control variable, with infinite values of corresponding to jumps in m. Our
approach of viewing m as a control variable parallels our treatment of the agent’s self-insurance problem(in which consumption, which has the same units as m, is a control variable), and facilitates connections
to discrete-time reporting problems (see the discussion below and in Section 7.3).
47The details are analogous to those from our derivation of the agent’s self-insurance solution(Lemma 5.1) in Appendix E. The fact that V DR satisfies [6.8] with V̂ DR † 0 (rather than V̂ DR “ 0)follows for Lemma E.1, with [NP-m] replacing [NP-A]. That V DR satisfies [6.9] follows from standard
results in stochastic control (Yong and Zhou 1999, Theorem 3.3; Touzi 2018, Propositions 2.4-2.5).
that the agent’s unique
48Plugging [6.8] into the first-order condition from [6.9]” yields´¯ı maximizer at
v
‹
v
DR
´1
‹
v
state pAt , bt q, call it m pAt , bt q, satisfies V̂ “ ´r exp ✓ Āpr; q ` r` m pAt , bt q . Plugging this`˘

expression back into [6.9] yields that V̂ DR “ ´r´1 exp ✓Āpr; q (which equals V̂ SI from [5.10]) andthat m‹ pAvt , bt q “ 0, as desired. One can then verify that the implied strategy is optimal by following thesame verification argument from the proof of Lemma 5.1 (see Appendix E).

33

mt “ 0) even at off-path histories where he has recently misreported (lim"Œ0 mt´" ‰ 0),

requiring him to submit a discontinuous “jump” report. This implication is naturalfrom two perspectives. First, in terms of the indirect implementation, it simply meansthat the agent immediately “re-initializes” his actual consumption cm
t at the optimalv
level ĈpAt , bt q prescribed by Lemma 5.1. Second, it is well known that in discrete-time“Markovian” contracting problems, a contract is IC if and only if it satisfies the (seemingly
stronger) property that the agent finds it optimal to truthfully report at all histories, bothon- and off-path. 49 The strategy derived here exhibits the continuous-time version ofthis property, highlighting a close connection between discrete- and continuous-time
reporting problems that we expect is valid beyond PPI’s model (see Section 7.3 below).
�. Discussion

We conclude by discussing implications of our analysis for the fully optimal contract
in PPI’s hidden endowment model, long-run properties of optimal insurance contracts,
and the relation between discrete- and continuous-time contracting models.
7.1. Fully Optimal Contracts
Our analysis leaves open an important question: What is the fully optimal contract in
PPI’s hidden endowment model? The answer depends on whether endowment shocks
are transient ( ° 0) or permanent ( “ 0).

Transient Shocks. When ° 0, it is natural to ask whether the optimal SI Contract
is fully optimal. Theorem 7 in Appendix J suggests a negative answer: under regularity
conditions, the optimal SI Contract is strictly suboptimal within the broader class of“FO-IC contracts,” i.e., the class of contracts for which the agent’s value function admits
an envelope formula. 50 If the optimal FO-IC contract is actually IC (i.e., the first-orderapproach is valid), it follows that the optimal SI Contract is not fully optimal. This
is to be expected: SI Contracts are renegotiation-proof and “stationary” (Section 5.4),
whereas prior work on contracting with persistent states has found that fully optimal
contracts typically violate both properties (e.g., Fernandes and Phelan 2000).
A full solution to PPI’s model with transient shocks remains an important open
question. While we do not know the answer, our analysis does have some implications
49In the special case of i.i.d. types, this fact forms the basis of the classic recursive formulations of
the agent’s incentive constraints in terms of promised utility (Green 1987; Thomas and Worrall 1990;
Atkeson and Lucas 1992). For settings with persistent private information, see Fernandes and Phelan(2000, Lemma 2.1, Theorem 2.1) for an early treatment and Pavan, Segal, and Toikka (2014, pp. 620-22,
645-46) for a modern treatment and general definition of “Markovian” environments. This property does
not appear in PPI because it is ruled out by PPI’s IML assumption, as discussed in Section 4.1.
50See Appendix A for a formal definition. As discussed there, this class includes all contracts consideredin PPI’s analysis, and we conjecture that it includes all IC contracts.

34

for it. We record them in Appendix A with the hope of stimulating work on this problem.
Permanent Shocks. When “ 0, Theorem 1 shows that Contract PPI is optimal
among SI Contracts. For this special case, Theorem 6 in Appendix J further shows thatContract PPI is optimal among all FO-IC contracts. This confirms PPI’s results for the“ 0 case (though we offer a simpler proof). The intuition mirrors that of Theorem 1(ii):when “ 0, the principal cannot manipulate his risk exposure, and therefore focuses on
eliminating drift distortions. What is potentially surprising is that this logic extends tothe full class of FO-IC contracts.
In a more general setting, Bloedel, Krishna, and Strulovici (2022) provide an explanation by observing that, under permanent shocks, the agent is necessarily indifferent
among (essentially) all reporting strategies under any FO-IC contract. Roughly speaking,when the agent’s type is subject to permanent shocks, his information rents are “so large”
that the principal cannot elicit any useful information. By contrast, when the agent’s
type is mean-reverting—even arbitrarily slowly—nontrivial screening and risk-sharing
is possible because the agent’s private information is “short-lived.” 51 We conclude thatthe nature of IC constraints under permanent shocks are, at least in some respects,
fundamentally different than under transient shocks (even as Œ 0). It is therefore notsurprising that optimal contracts differ in these two cases, as well.
7.2. Immiseration and Persistence
Perhaps the most striking claim in PPI is that the classic immiseration property of (fully)
optimal insurance contracts breaks down in PPI’s model due to (i) the persistence of theagent’s information and (ii) fundamental differences between IC constraints in discreteand continuous-time models. Our analysis demonstrates that this claim does not follow
from PPI’s results in the generic case of transient shocks ( ° 0) because Contract PPI
is optimal only when shocks are permanent ( “ 0). We address the role of persistence
here and turn to the role of continuous time in Section 7.3 below.
Transient Shocks. Theorem 3 in Appendix G shows that the optimal SI Contract
generates immiseration when persistence is low ( ° ) and bliss when persistence ishigh ( † ). Because SI Contracts are not fully optimal when ° 0 (under regularity
conditions), this finding does not have direct implications for the fully optimal contract.
In a general discrete-time setting, Bloedel, Krishna, and Leukhina (2021) find thatfully optimal contracts generate immiseration when the agent’s private type follows afinite-state, fully-connected Markov process. As their model permits arbitrarily good
approximations of PPI’s hidden endowment model, it is natural to expect that the fully
51For any ° 0, the influence of the agent’s time-t type has a vanishing influence on his time-⌧ type as⌧ Ñ 8 (recall Footnote 10), so that the principal and agent have “almost symmetric” information over
the agent’s time-⌧ preferences.

35

optimal contract in PPI’s model may also generate immiseration for all ° 0. Further
investigation of this question is an important direction for future research.
Permanent Shocks. When “ 0, immiseration fails under the optimal FO-IC contract(Contract PPI). We view this as a knife-edge result specific to “ 0. The literature often
attributes immiseration to the principal’s manipulation of the agent’s risk exposure forcost-smoothing purposes (see BKL for details). Lemma J.1 in Appendix J formalizes asense in which the principal can manipulate the agent’s risk exposure if and only if ° 0.
Thus, when “ 0, the classic rationale for immiseration is shut off by construction. 52
7.3. Continuous vs. Discrete Time
A central claim in PPI is that (i) there is a fundamental difference between reporting
incentives in discrete- and continuous-time models and (ii) this difference is at least
partially responsible for the differences between Contract PPI and optimal contracts
derived in the prior literature. This claim is based on the observation that, in PPI’s model,
the agent’s time-t choice of t affects his future misreports m⌧ and actual consumptionmcm⌧ for ⌧ ° t, but does not affect the current misreport mt or consumption ct , whichapparently stands in contrast to discrete-time models in which the agent can freely choose
his current misreport and therefore affect his current consumption. 53 While the latter
observation is correct, our analysis supports neither component of PPI’s claim.
(i) Reporting Incentives. As discussed in Section 6.4, our direct verification of IC
for DR-SICs suggests fundamental similarities between reporting incentives in discreteand continuous-time models. We showed there how to “extend” the agent’s strategy
space (and the contract’s responses) to allow for discontinuous “jump” reports. In theextended problem, as in discrete-time models, the agent directly chooses his misreport
mt at time t and instantaneously affects his time-t consumption cm
t . We also described
why considering such “extensions” is without loss of generality for the purposes ofverifying IC, and why the agent’s incentive to truthfully report his current type evenat off-path histories—familiar from discrete-time models—makes such “extensions”
particularly natural in continuous time.
52In a more general setting with permanent shocks, Bloedel, Krishna, and Strulovici (2022) show thatthe optimal long-run properties depend on details of the agent’s type process and utility function.
53For instance, PPI writes (p. 1235): “[T]hese differences [between Contract PPI and the optimal
contract in Thomas and Worrall (1990)] rely at least partly on differences in the environments. In thediscrete [time] analogue of my model, when deciding what to report in the current period, the agent
trades off current consumption and future promised utility. In my continuous-time formulation, the agent’s
private state follows a process with continuous paths and the principal knows this. Thus in the current
period the agent only influences the future increments of the reported state. Thus current consumption is
independent of the current report and all that matters for the reporting choice is how future transfers areaffected . . . [T]he reporting problem and, hence, the incentive constraints become fully forward-looking. . . ” Similar statements are made elsewhere in PPI (pp. 1239, 1257-58, 1263, 1264).

36

In Appendix I, we show how to directly analyze the agent’s reporting problem in
DR-SICs while maintaining PPI’s original assumption that misreports have absolutely
continuous sample paths (m P M). Under this assumption, the agent can still approximate
the effect of a jump report at time t through a sequence of strategies in M by sending| ⌧ | Ñ 8 for a vanishing interval of times ⌧ ° t. Intuitively, although the agent can nolonger instantaneously affect his misreport mt or his consumption cm
t , he can still affectthese variables at arbitrarily close times ⌧ ° t, and finds it optimal to do so because
DR-SICs respond similarly to “jump” reports and such “approximate jump” reports (seeAppendix B). While the details of this argument are specific to PPI’s model and the classof DR-SICs, we expect that the basic points apply more broadly in other continuous-time
contracting models. Consequently, our analysis suggests that the differences between
discrete- and continuous-time models identified in PPI (see Footnote 53) are cosmetic,
rather than substantive. 54(ii) Optimal Contracts. Our results extend to the natural discrete-time analogue of
PPI’s hidden endowment model in which the agent’s endowment follows a Gaussian
AR(1) process. Observation 1 plainly extends. For Observations 2–3 and the associatedanalysis of SI Contracts, recall that our solution to the agent’s self-insurance problem(Lemma 5.1) is the continuous-time limit of Caballero’s (1991) and Wang’s (2003)
discrete-time self-insurance solutions. Thus, we can replicate our analysis of SI Contracts
and DR-SICs in discrete time, using those papers’ solutions to construct the discrete-time
SI Contracts and defining the discrete-time version of Contract PPI as the particular
one with zero taxes. Discrete-time versions of Observations 2–3 and Theorems 1 and 2
readily follow. Consequently, any differences between Contract PPI and the optimal
contracts in the discrete-time models on which PPI’s model is based (Thomas and Worrall
1990; Atkeson and Lucas 1992) are not driven by the distinction between discrete andcontinuous time. Rather, the discrete-time analogue of Contract PPI is also genericallysuboptimal in the discrete-time version of PPI’s model. 55
Appendix

Appendices A–B are presented here. Appendices C–K are in the Online Appendix.
A. PPI’s Derivation of Contract PPI

We begin with a few (sometimes implicit) definitions from PPI, stated in our terminology.
By the Martingale Representation Theorem, the agent’s promised utility process (defined
54Future work might explore more general connections between discrete- and continuous-time screening
problems by studying discrete-time models with vanishing period length (cf. Sadzik and Stacchetti 2015).
55Our finding that Contract PPI is optimal in an alternative model with hidden savings extends todiscrete time by the same logic described above. Our confirmation that Contract PPI is optimal underpermanent shocks extends to discrete time per a special case of Bloedel, Krishna, and Strulovici (2022).

37

in [3.1]) under any contract satisfies[A.1]

dqt “ p⇢qt ´ ut q dt `t

dWty ,

where W y is the “inferred shock process” under reporting strategy y (as defined in
Section 3) and is a y -adapted sensitivity process determined by the contract. Recall
the agent’s marginal promised utility process p from [3.2]. The condition[FO-IC]t ` pt ” 0.

corresponds to the familiar envelope formula for the agent’s information rents: it equates
the marginal value t of an increase in the agent’s report and the marginal value ´ptof an increase in the agent’s true type. Any contract that satisfies [FO-IC] is said to be
first-order IC (FO-IC). The class of FO-IC contracts includes all DR-SICs (which satisfy[FO-IC] with pt ” f pr; qqt ) and all other contracts allowed for in PPI’s treatment. 56 , 57
We say that a contract is an optimal FO-IC contract if it minimizes the principal’s lifetime
cost under truthtelling (as in [2.2]) among all FO-IC contracts delivering a pre-specified
promised utility level. Note that an FO-IC contract, including the optimal one, need not
be IC.
Standing Assumption. For the remainder of this appendix, we focus on the genericcase of PPI’s hidden endowment model with transient shocks ( ° 0).

PPI’s Claims. PPI’s claim that Contract PPI is the optimal M§LAC -IC contract can be
decomposed into three sub-claims:
56§3 of PPI claims (in a general finite-horizon setting) that all IC contracts are FO-IC, and we conjecturethat this is indeed the case in PPI’s hidden endowment model. The basic idea is to show that [FO-IC] is
implied by [IC] by deriving the former as an infinitesimal optimality condition for the agent that rules out“local” deviations from truthtelling. Technically, this requires appealing to the envelope theorem (e.g.,
Kapi ka 2013; Pavan, Segal, and Toikka 2014), the stochastic maximum principle (as in PPI, pp. 1243-44,1264), or other variational arguments to establish that such infinitesimal conditions are well-defined.
The usual approach in the literature is to allow for all conceivable contracts while imposing regularity
conditions on the agent’s type process and preferences to ensure that such arguments are applicable(e.g., Pavan, Segal, and Toikka 2014, Theorem 1). However, to our knowledge, the agent’s CARA utility
function in the present model does not satisfy regularity conditions found in the literature because it
is unbounded below and does not satisfy standard growth conditions. To avoid these purely technicalconsiderations, we simply restrict attention to the class of FO-IC contracts.
57PPI initially states a weaker version of [FO-IC] requiring only that t ` pt • 0 (display (10) on p.
1244), which is the appropriate “one-sided” envelope formula when either NHB (m § 0) or IML ( § 0)
is imposed. However, PPI’s entire analysis of optimal contracts (§§5-8) is based on the version of [FO-IC]
stated above. In §5, PPI writes (p. 1250): “We now assume that the incentive constraint . . . binds, so that
“ ´p. We relax this condition in an example below and verify that it holds. However, we also conjecturethat it holds more generally.” PPI does not carry out the asserted verification, instead relying on “ ´pthroughout §§5-8. Nonetheless, at least in the special case of permanent shocks ( “ 0) this restriction isindeed without loss of optimality (Remark J.2 in Appendix J.2).

38

• Claim 1: Contract PPI is the optimal FO-IC contract.
• Claim 2: Contract PPI is M§LAC -IC.
• Claim 3: The optimal M§LAC -IC contract is FO-IC.
Below, we identify issues with each of these claims and where they arise in PPI. 58
PPI’s Argument for Claim 1. §5 of PPI presents an HJB equation for the principal’svalue function over [FO-IC] contracts in the infinite-horizon setting (see display (19)on p. 1249 and the first-order conditions in displays (24)–(25) on p. 1251). PPI thenpresents the main derivation of Contract PPI in §6.2.1 and §6.2.2 (pp. 1252-54) and§A.3.1 (pp. 1269-71). The main steps of that derivation are as follows:
Step 1. Conjecture that the principal’s value function at time t—which in general can be
written as a function Jpyt , qt , pt q—depends on marginal promised utility pt (as definedin [3.2]) only through the ratio kt “ pt {qt . Also conjecture that the dependence on kt is
ˆ t , qt q ` hpkt q for some functions Jˆ and h (seeadditively separable, viz., Jpyt , qt , pt q “ Jpydisplay (26) on p. 1252).
Step 2. Assume that hp¨q is smooth, plug the conjectured form of the value function from
Step 1 into the principal’s HJB equation. This delivers a second-order ODE for hp¨q (seedisplay (A.11) on p. 1269).
Step 3. Use the policy functions derived from the HJB equation, together with Itô’slemma, to conclude that the resulting k process must satisfy a particular law of motion(display (28) on p. 1253).
Step 4. Numerically solve the ODE for hp¨q derived in Step 2, plot the solution for specificparameter values, graphically observe that hp¨q appears to be minimized at the value
k0˚ “ ✓ {p⇢ ` q, and conclude that k0˚ is the optimal initial condition for the k process(pp. 1270-71). (Recall that the initial conditions q0 and y0 “ b0 are given, while theprincipal’s choice of FO-IC contract determines p0 .)Step 5. Observe that, given the law of motion from Step 3 and the initial condition k0˚from Step 4, the k process is necessarily constant. Conclude from this that the policy
function from the HJB generates Contract PPI, and therefore that Contract PPI is the
optimal [FO-IC] contract (see pp. 1253-54).
Issues with Claim 1. In Appendix J (see Remark J.10), we verify that Steps 1–3 andStep 5 above are correct as stated (while also showing that PPI’s conjectures in Step 1
can be derived from first principles, and making explicit some technical assumptions
that PPI makes implicitly). Meanwhile, Theorem 1 establishes the existence of an FO-IC
58Recall from Section 4.2 that PPI does not fully specify the agent’s strategy space in the infinite-horizon
setting of §§5–8, so we adopt our best understanding of PPI’s assumptions, which is that the agent’s
LAC . Claim 1 and the issues that we identify with it are independent of this convention,
feasible set is F “ M§as the class of FO-IC contracts is defined independently of the agent’s feasible set.

39

contract that strictly dominates Contract PPI. 59 It follows that PPI’s conclusion in Step
4—that k0˚ is the optimal initial condition—is incorrect. We also note that PPI also does
not specify boundary conditions for the ODE for hp¨q in Step 4; it is therefore unclear
what ODE system the numerical solution in Step 4 pertains to.
Issues with Claims 2–3. These follow from discussion above:• In Section 4.2, we describe issues with PPI’s argument for Claim 2. We also show that
Claim 2 is false by constructing a strategy m P M§LAC that the agent strictly prefers totruthtelling (Observation 2).
• In Footnotes 56 and 57 above, we describe how PPI (i) initially presents a weaker“one-sided” version of [FO-IC], (ii) claims that it is satisfed by every M§LAC -IC contract
in the finite-horizon setting, (iii) restricts attention to FO-IC contracts (as definedhere) when studying optimal contracts in the infinite-horizon setting, and (iv) does
not actually verify that this restriction is without loss of optimality. Consequently,
PPI does not establish Claim 3. We nonetheless conjecture that Claim 3 is true.
Implications for the Optimal FO-IC Contract. Through Steps 1–3 above, PPI derives two correct properties of the optimal FO-IC contract. 60 Taking b0 and q0 as given,
let k0: P R`` denote the true optimal initial condition for the process kt :“ pt {qt . First,
PPI finds that the principal’s optimal lifetime cost is[A.2]

Jpb0 , q0 , k0: q0 q “ J ˚ pb0 , q0 q `

2

2⇢2 ✓

´ ¯2
k0: ,

where J ˚ pb0 , q0 q is the first-best value function (recall Footnote 35). Second, PPI findsthat the agent’s initial recommended consumption is[A.3]

c:0 “ cpq0 , ⇢q,where cpq0 , ⇢q “ ´ logp´⇢q0 q{✓. Note that Contract PPI specifies the same initial condition for recommended consumption. Taken together, our results and [A.2]–[A.3] have thefollowing implications for the optimal FO-IC contract: 61
1. The initial condition satisfies k0: † k0˚ , where k0˚ “ ⇢✓{p⇢ ` q is the initial condition
for Contract PPI. This holds because the principal’s cost of Contract PPI (stated in
2
2[5.13]) can be expressed as J ˚ pb0 , q0 q ` 2⇢2 ✓ pk0˚ q , our Theorem 1 and the fact that SI
59Theorem 2 shows that the contract we identify in Theorem 1 is IC in a suitable sense, but this is not
relevant for the present discussion of FO-IC contracts.
60See the calculations on pp. 1269–71 in §A.3.1 of PPI, which assume that hp¨q is twice continuously
differentiable and satisfies h2 pk0: q ° 0. We maintain these assumptions in the discussion here.
61We emphasize that our results, [A.2]–[A.3], and the implications below are mutually consistentbecause our DR-SICs correspond to the strict subset of FO-IC contracts that are “stationary” (viz., feature
a constant kt “ pt {qt process). See Section 5.4, and Lemma G.1 in Appendix G.

40

Contracts are FO-IC imply that this cost is strictly greater than Jpb0 , q0 , k0: q0 q, thekt “ pt {qt process is strictly positive, and hence [A.2] implies k0˚ ° k0: .
“‰2. At t “ 0, we have dkt “ p⇢ ` q k0: ´ k0˚ dt † 0. The expression for dk0 follows fromsubstituting h1 pk0: q “ 0 into the expressions for ĉpk0: q and Q̂pk0: q on pp. 1269–70 of
PPI, and then substituting those expressions into display (28) on p. 1253 of PPI. The
strict inequality follows from the first implication above.
Further analysis of the optimal FO-IC contract is an exciting direction for future research.
B. On the Agent’s Reporting Problem in DR-SICs

When the agent’s feasible set is F Ñ M, his reporting problem can be cast as oneof stochastic control with states pAvt , yt , mt q and control t , as in PPI. For a DR-SIC
with rate r ° 0, V NJ pAvt , yt , mt q denotes the agent’s value function under the “no jump”
feasible set F “ Mr ; as in Section 6.4, V DR pAvt , yt ´ mt q denotes that under the extendedrfeasible set F “ Mext
(where we have used the identity bt ” yt ´ mt ). We show thatthese two value functions coincide, i.e., the agent does not strictly benefit from having
access to jump reports (as in the treatment of IC from Section 6) either on- or off-path.
(In Appendix I, we directly analyze the “no jump” reporting problem.)Lemma B.1. For any DR-SIC with rate r ° 0, the agent’s value functions satisfy[B.1]

V DR pAvt , yt ´ mt q ” V NJ pAvt , yt , mt q ” qt exp rf pr; qmt s .

Proof. By construction, qt ” V SI pAvt , yt q. Thus, [5.10], [6.8], and the identity mt ”r
yt ´ bt yield V DR pAvt , yt ´ mt q ” qt exp rf pr; qmt s. Next, Mr à Mext
implies that
NJv
V pAt , yt , mt q § qt exp rf pr; qmt s. If mt “ 0, the upper bound is trivially attained; formt ‰ 0, we show its attainment via an approximation argument. Let pAvt , yt , mt q withmt ‰ 0 be a state at time t in the agent’s reporting problem with (non-extended) feasible
set Mr . For any continuation strategy m P Mr and time T ° t, integrating [A.1] (withu⌧ ” rq⌧ and ⌧ ” ´f pr; qq⌧ ) over ⌧ P rt, T s delivers
[B.2]

„

ˆ
˙⇢
ªT
qT “ q̂T ¨ exp f pr; q pmt ´ mT q ` mt pT ´ tq ´
m⌧ d⌧
,
t

where q̂T is the counterfactual time-T promised utility that would have arisen under
the same endowment shocks pW⌧ q⌧ Prt,T s had the agent set ⌧ ” 0 on ⌧ P rt, T s. 62
62Fix the path pm⌧ q⌧ Pr0,tq . Under the continuation strategy m, the “inferred shock process” satisfies≥Tm⌧ d⌧ . Under the continuation strategy ⌧ ” 0 for all ⌧ P rt, T s, the0≥tcorresponding process, call it Ŵ y , satisfies ŴTy “ W T ` mt ` 0 m⌧ d⌧ ` mt pT ´ tq. IntegratingSIv[A.1] (with u⌧ ” rq⌧ and ⌧ ” ´f pr; qq⌧ ) from thecondition” fixed initial´¯ı qt “ V pAt , yt ´ mt q overyy
⌧ P rt, T s under both strategies yields qT “ q̂T exp f pr; q ŴT ´ WT , which reduces to [B.2].
WTy “

W T ` mT `41

For each T ° t, consider the specific continuation strategy mT P Mr induced byT
T⌧ :“ ´mt {pT ´ tq ¨ 1p⌧ P rt, T sq. Under m , [B.2] reduces to
[B.3]

„

⇢
pT ´ tqqT “ q̂T ¨ exp rf pr; qmt s ¨ exp ´ f pr; qmt ¨.
2

As T Œ t, mT approximates an instantaneous jump to mt` “ 0 and truthful reporting forall ⌧ ° t (the optimal continuation strategy from the extended problem in Section 6.4);
we have q̂T Ñ q̂t “ qt and thus qT Ñ qt exp rf pr; qmt s P-a.s. Furthermore, because mT
specifies truthful reporting for ⌧ • T , qT is also the agent’s true time-T continuation
utility. It follows that V NJ pAvt , yt , mt q • qt exp rf pr; qmt s, as desired.
Remark B.2. Lemma B.1 and its proof have a few notable implications:(i) Recommended consumption satisfies ct ” cpqt , rq, so sending T Œ t in [B.3] also implies that the agent’s recommended and actual consumption (hence, also the transfersst ) converge to those under the optimal extended strategy from Section 6.4.
(ii) Because V NJ “ V DR and the latter value function is uniquely attained by the optimal“report truthfully at all histories” strategy from Section 6.4—which is not in Mr —it
follows that when the agent’s feasible set is Mr , he does not have a well-defined
optimal continuation strategy at off-path histories where mt ‰ 0. At such histories,
the only way to attain continuation value V DR pAvt , yt ´ mt q is to approximate a jumpreport back to mt “ 0, as in the above proof.
(iii) By restricting attention to mt § 0, the arguments here and in Section 6.4 apply
almost verbatim when NHB is imposed, i.e., the agent’s feasible set is M´r . Given thisfeasible set, the agent’s value function is therefore V NHB pq, mq “ q exp rf pr; qms
as in [B.1], but restricted to the domain where m § 0. As noted in Section 4.1, forContract PPI (r “ ⇢), this differs from the agent’s value function when the stronger[IML] constraint is imposed (cf. [4.1]).
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